Research Article Empirical Likelihood for Partial...

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Research Article Empirical Likelihood for Partial Parameters in ARMA Models with Infinite Variance Jinyu Li, 1 Wei Liang, 2 and Shuyuan He 3 1 School of Sciences, China University of Mining and Technology, Xuzhou 221116, China 2 School of Mathematical Sciences, Xiamen University, Xiamen 361005, China 3 School of Mathematical Sciences, Capital Normal University, Beijing 100048, China Correspondence should be addressed to Jinyu Li; [email protected] Received 31 March 2014; Accepted 24 August 2014; Published 2 September 2014 Academic Editor: Charalampos Tsitouras Copyright © 2014 Jinyu Li et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper proposes a profile empirical likelihood for the partial parameters in ARMA(, ) models with infinite variance. We introduce a smoothed empirical log-likelihood ratio statistic. Also, the paper proves a nonparametric version of Wilks’s theorem. Furthermore, we conduct a simulation to illustrate the performance of the proposed method. 1. Introduction Consider the stationary ARMA(, ) time series { } gener- ated by = 1 −1 +⋅⋅⋅+ − + + 1 −1 +⋅⋅⋅+ − , (1) where the innovation process { } is a sequence of i.i.d. random variables. When ( 2 )=∞, model (1) is an infinite variance autoregressive moving average (IVARMA) model, which defines a heavy-tailed process { }. For model (1), statistical inference has been explored in many studies (see, e.g., [1, 2]). Recently, for example, Pan et al. [3] and Zhu and Ling [4] proposed a weighted least absolute deviations esti- mator (WLADE) for model (1) and obtained the asymptotic normality. However, in the building of ARMA models, we are usually only interested in statistical inference for partial parameters. For example, in the sparse coefficient (a part of zero coef- ficients) ARMA models, it is necessary to determine which coefficient is zero. For model (1), one traditional method is to construct confidence regions for the partial parameters of interest by normal approximation as in [3]. However, since the limit distribution depends on the unknown nuisance parameters and density function of the errors, estimating the asymptotic variance is not a trivial task. Based on these, this paper tries to put forward a new method for the esti- mation of partial parameters of ARMA models. We propose an empirical likelihood method, which was introduced by Owen [5, 6]. Based on the estimating equations of WLADE, a smoothed profile empirical likelihood ratio statistic is derived, and a nonparametric version of Wilks’s theorem is proved. erefore, we can construct confidence regions for the partial parameters of interest. Also, simulations suggest that, for relative small sample cases, the empirical likelihood confidence regions are more accurate than those confidence regions constructed by the normal approximation based on the WLADE proposed by Pan et al. [3]. As an effective nonparametric inference method, the empirical likelihood method produces confidence regions whose shape and orientation are determined entirely by the data and therefore avoids secondary estimation. In the past two decades, the empirical likelihood method has been extended to many applications [7]. ere are also many studies of empirical likelihood method for autoregressive models. Monti [8] considered the empirical likelihood in the frequency domain; Chuang and Chan [9] developed the empirical likelihood for unstable autoregressive models with innovations being a martingale difference sequence with finite variance; Chan et al. [10] applied the empirical Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 868970, 10 pages http://dx.doi.org/10.1155/2014/868970

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Research ArticleEmpirical Likelihood for Partial Parameters in ARMA Modelswith Infinite Variance

Jinyu Li1 Wei Liang2 and Shuyuan He3

1 School of Sciences China University of Mining and Technology Xuzhou 221116 China2 School of Mathematical Sciences Xiamen University Xiamen 361005 China3 School of Mathematical Sciences Capital Normal University Beijing 100048 China

Correspondence should be addressed to Jinyu Li ljycumt163com

Received 31 March 2014 Accepted 24 August 2014 Published 2 September 2014

Academic Editor Charalampos Tsitouras

Copyright copy 2014 Jinyu Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper proposes a profile empirical likelihood for the partial parameters in ARMA(119901 119902) models with infinite variance Weintroduce a smoothed empirical log-likelihood ratio statistic Also the paper proves a nonparametric version of Wilksrsquos theoremFurthermore we conduct a simulation to illustrate the performance of the proposed method

1 Introduction

Consider the stationary ARMA(119901 119902) time series 119910119905 gener-

ated by

119910119905= 1205931119910119905minus1

+ sdot sdot sdot + 120593119901119910119905minus119901

+ 120576119905+ 1205991120576119905minus1

+ sdot sdot sdot + 120599119902120576119905minus119902

(1)

where the innovation process 120576119905 is a sequence of iid

random variables When 119864(1205762119905) = infin model (1) is an infinite

variance autoregressive moving average (IVARMA) modelwhich defines a heavy-tailed process 119910

119905 For model (1)

statistical inference has been explored in many studies (seeeg [1 2]) Recently for example Pan et al [3] and Zhu andLing [4] proposed a weighted least absolute deviations esti-mator (WLADE) for model (1) and obtained the asymptoticnormality

However in the building ofARMAmodels we are usuallyonly interested in statistical inference for partial parametersFor example in the sparse coefficient (a part of zero coef-ficients) ARMA models it is necessary to determine whichcoefficient is zero For model (1) one traditional method isto construct confidence regions for the partial parameters ofinterest by normal approximation as in [3] However sincethe limit distribution depends on the unknown nuisanceparameters and density function of the errors estimating

the asymptotic variance is not a trivial task Based on thesethis paper tries to put forward a new method for the esti-mation of partial parameters of ARMA models We proposean empirical likelihood method which was introduced byOwen [5 6] Based on the estimating equations of WLADEa smoothed profile empirical likelihood ratio statistic isderived and a nonparametric version of Wilksrsquos theorem isproved Therefore we can construct confidence regions forthe partial parameters of interest Also simulations suggestthat for relative small sample cases the empirical likelihoodconfidence regions are more accurate than those confidenceregions constructed by the normal approximation based onthe WLADE proposed by Pan et al [3]

As an effective nonparametric inference method theempirical likelihood method produces confidence regionswhose shape and orientation are determined entirely bythe data and therefore avoids secondary estimation In thepast two decades the empirical likelihood method has beenextended to many applications [7] There are also manystudies of empirical likelihood method for autoregressivemodels Monti [8] considered the empirical likelihood inthe frequency domain Chuang and Chan [9] developedthe empirical likelihood for unstable autoregressive modelswith innovations being a martingale difference sequencewith finite variance Chan et al [10] applied the empirical

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 868970 10 pageshttpdxdoiorg1011552014868970

2 Journal of Applied Mathematics

likelihood to near unit root AR(1) model with infinite vari-ance errors Li et al [11 12] respectively used the empiricallikelihood to infinite variance AR(119901) models and model (1)

The rest of the paper is organized as follows In Section 2we propose the profile empirical likelihood for the parametersof interest and show the main result Section 3 provides theproofs of the main results Some simulations are conductedin Section 4 to illustrate our approach Conclusions are givenin Section 5

2 Methodology and Main Results

First the parameter space is denoted by Θ sub 119877119901+119902 whichcontains the true value 120579

0of the parameter 120579 as an inner point

For 120579 = (1205931 120593

119901 1205991 120599

119902) put

120576119905(120579) =

119910119905minus

119901

sum119894=1

120593119894119910119905minus119894

minus

119902

sum119895=1

120599119895120576119905minus119895

(120579) if 119905 gt 0

0 if 119905 le 0

(2)

where 119910119905equiv 0 for all 119905 le 0 and note that 120576

119905(1205790) = 120576119905 because

of this truncationWe define the objective function as

119878119899(120579) =

119899

sum119905=119906+1

119908119905

1003816100381610038161003816120576119905 (120579)1003816100381610038161003816 (3)

where 119906 ge max(119901 119902) and the weight function 119908119905= 1(1+

sum119905minus1

119896=1119896minus120572|119910119905minus119896|)4 depending on a constant 120572 gt 2 The

WLADE denoted by 120579 is a lacol minimizer of 119878119899(120579) in

a neighborhood of 1205790[3] Denote 119860

119905(120579) = (119860

1199051(120579)

119860119905119901+119902

(120579))120591 where119860

119905119894(120579) = minus120597120576

119905(120579)120597120579

119894 By (8119) of Brock-

well and Davis [13] it holds for 119905 gt max(119901 119902) that

119860119905119894(120579) +

119902

sum119895=1

120599119895119860119905minus119895119894

(120579) = 119910119905minus119894 119894 = 1 119901

119860119905119894+119901

(120579) +

119902

sum119895=1

120599119895119860119905minus119895119894+119901

(120579) = 120576119905minus119894(120579) 119894 = 1 119902

(4)

Hence 120579 satisfies estimating equation

1

119899 minus 119906

119899

sum119905=119906+1

119908119905119860119905(120579) sgn 120576

119905(120579) = 0 (5)

where sgn(119909) = minus1 for 119909 lt 0 and =1 for 119909 ge 0

(see [14]) Note that the above estimating equation is notdifferentiable at point 120579 such that 120576

119905(120579) = 0 for some 119905

This causes some problems for our subsequent asymptoticanalysis To overcome this problem we replace it with asmooth function Define a probability density kernel119870(sdot) [15]such that int+infin

minusinfin119909119895119870(119909)119889119909 = 0 120581 for 119895 = 1 2 respectively

where 120581 = 0 Let 119866ℎ(119909) = int

119909ℎ

minus119909ℎ119870(119906)119889119906 for ℎ gt 0 Then a

smoothed version of (5) is

1

119899 minus 119906

119899

sum119905=119906+1

119908119905119860119905(120579) 119866ℎ(120576119905(120579)) = 0 (6)

Let 119898119905ℎ(120579) = 119908

119905119860119905(120579)119866ℎ(120576119905(120579)) a smoothed empirical

log-likelihood ratio is defined as

119897ℎ(120579)

= minus2 sup119899

sum119905=119906+1

log ((119899 minus 119906) 119901119905) |

119899

sum119905=119906+1

119901119905119898119905ℎ(120579) = 0

119901119905ge 0

119899

sum119905=119906+1

119901119905= 1

(7)

Using the Lagrange multiplier the optimal value of 119901119905is

derived to be

119901119905(120579) =

1

(119899 minus 119906) (1 + 120582(120579)120591119898119905ℎ(120579))

119906 + 1 le 119905 le 119899

(8)

where 120582(120579) is a 119901 + 119902-dimensional vector of Lagrangemultipliers satisfying

1

119899 minus 119906

119899

sum119905=119906+1

119898119905ℎ(120579)

1 + 120582(120579)120591119898119905ℎ(120579)

= 0 (9)

This gives the smoothed empirical log-likelihood ratio statis-tic

119897ℎ(120579) = 2

119899

sum119905=119906+1

log (1 + 120582(120579)120591119898119905ℎ(120579)) (10)

Let 120579 = (120601120591 120596120591)120591 where 120596 isin 119877119898 (1 le 119898 le 119901 + 119902)

is the parameter of interest and 120601 isin 119877119901+119902minus119898 is the nuisanceparameter Note that 119898 = 119901 + 119902 means no nuisance param-eters Let 120601

0and 120596

0denote the true values of 120601 and 120596 respec-

tively The profile empirical likelihood is defined as

119897119901(120596) = min

120601

119897ℎ(120601 120596) (11)

That is 119897119901(120596) = 119897

ℎ(120601(120596) 120596) where 120601 = 120601(120596) = argmin

120601119897ℎ

(120601 120596)The following conditions are in order

(A1) The characteristic polynomial 120601(119911) = 1 minus 1205931119911 minus sdot sdot sdot minus

120593119901119911119901 and 120579(119911) = 1+120599

1119911+ sdot sdot sdot + 120599

119902119911119902 have no common

zeros and all roots of 120601(119911) and 120579(119911) are outside theunit circle

(A2) The innovation 120576119905 has zero median and a differen-

tiable density 119891(119909) satisfying the conditions 119891(0) gt 0sup119909isin119877

|119891(119909)| lt 1198611lt infin and sup

119909isin119877|1198911015840(119909)| lt 119861

2lt

infin Furthermore 119864|120576119905|120575 lt infin for some 120575 gt 0 and

120572 gt max2 2120575(A3) As 119899 rarr infin 119906 rarr infin and 119906119899 rarr 0(A4) The second derivative of 119870 exists in 119877 and 1198701015840(119909) and

11987010158401015840(119909) are bounded(A5) ℎ = 1119899120574 with 14 lt 120574 lt 13

First we show the existence and consistency of 120601(1205960)

Journal of Applied Mathematics 3

Proposition 1 Let 119889119899= 1119899120573 with max13 31205742 lt 120573 lt

12 Assume (A1)ndash(A5) hold then as 119899 rarr infin with probability1 there exists a local minimizer 120601 of 119897

ℎ(120601 1205960) which lies in the

interior of the ball 119861 = 120601 120601 minus 1206010 le 119889

119899 Moreover 120601 and

= 120582(120601 1205960) satisfy

1198761119899(120601 ) = 0 119876

2119899(120601 ) = 0 (12)

where

1198761119899(120601 120582) =

1

119899 minus 119906

119899

sum119905=119906+1

119898119905ℎ(120601 1205960)

1 + 120582120591119898119905ℎ(120601 1205960)

1198762119899(120601 120582)

=1

119899 minus 119906

119899

sum119905=119906+1

1

1 + 120582120591119898119905ℎ(120601 1205960)(120597119898119905ℎ(120601 1205960)

120597120601120591)

120591

120582

(13)

The following theorem presents the asymptotic distribu-tion of the profile empirical likelihood

Theorem 2 Under conditions of Proposition 1 as 119899 rarr infinthe random variable 119897

119901(1205960) with 120601 given in Proposition 1

converges in distribution to 1205942119898

If 119888 is chosen such that 119875(1205942119898le 119888) = 119886 then Theorem 2

implies that the asymptotic coverage probability of empiricallikelihood confidence region 119868

ℎ119888= (120596 119897

119901(120596) le 119888) will be 119886

that is 119875(1205960isin 119868ℎ119888) = 119875(119897

119901(1205960) le 119888) = 119886 + 119900(1) as 119899 rarr infin

3 Proofs of the Main Results

In the following sdot denotes the Euclidian norm for a vectoror matrix and 119862 denotes a positive constant which may bedifferent at different places For 119905 = 0 plusmn1 plusmn2 define

119880119905minus

119901

sum119894=1

1205930

119894119880119905minus119894

= 120576119905 119881

119905+

119902

sum119895=1

1205990

119895119881119905minus119895

= 120576119905 (14)

Put 119876119905

= (119880119905minus1 119880

119905minus119901 119881119905minus1 119881

119905minus119902)120591 119908119905

= 1(1 +

suminfin

119896=1119896minus120572|119910119905minus119896|)4 and the corresponding partial vector for 120601

0

is denoted by 1198761119905 Let

Σ = 119864 (119908119905119876119905119876120591

119905)

Σ1= 119864 (119908

119905119876119905119876120591

1119905)

Ω = 119864 (1199082

119905119876119905119876120591

119905)

(15)

Assumptions A1 and A2 imply that for 120575 = min(120575 1)

119864(

infin

sum119896=1

119896minus1205722 1003816100381610038161003816119910119905minus119896

1003816100381610038161003816)

120575

le

infin

sum119896=1

119896minus1205721205752

1198641003816100381610038161003816119910119905minus119896

1003816100381610038161003816120575

lt infin (16)

Hence suminfin119896=1

119896minus1205722|119910119905minus119896| lt infin with probability 1 which

ensures that 119908119905

is well defined Note that 119876119905 le

119862suminfin

119895=1119903119895|119910119905minus119895| for some 0 lt 119903 lt 1 and

11990812

119905

10038171003817100381710038171198761199051003817100381710038171003817 le

119862suminfin

119895=111990311989510038161003816100381610038161003816119910119905minus119895

10038161003816100381610038161003816

1 + suminfin

119896=1119896minus120572

1003816100381610038161003816119910119905minus1198961003816100381610038161003816le 119862

infin

sum119896=1

119903119896119896120572lt infin (17)

Then Σ Σ1 and Ω are well-defined (finite) matrices For

simplicity we denote (120601 1205960) and (120601

0 1205960) by 120601 and 120601

0

respectively in this section The following notations will beused in the proofs Let

119885119899(120601) = max

119906+1le119905le119899

1003817100381710038171003817119898119905ℎ (120601)1003817100381710038171003817

119876119899ℎ(120601) =

1

119899 minus 119906

119899

sum119905=119906+1

119898119905ℎ(120601)

119878 (120601) =1

119899 minus 119906

119899

sum119905=119906+1

119898119905ℎ(120601)119898

119905ℎ(120601)120591

(18)

To prove Proposition 1 we first prove the following lemmas

Lemma 3 Under the conditions of Proposition 1 as 119899 rarr infin

(119894) 119876119899ℎ(1206010) = 119874(radic

log 119899119899

) as

(119894119894)120597119876119899ℎ(1206010)

120597120601120591= minus2119891 (0) Σ

1+ 119900 (1) as

(19)

Proof of Lemma 3 For part (i) we may write

radic119899 minus 119906119876119899ℎ(1206010)

=1

radic119899 minus 119906

119899

sum119905=119906+1

119908119905119876119905119866ℎ(120576119905)

+1

radic119899 minus 119906

119899

sum119905=119906+1

[119908119905119876119905(119866ℎ(120576119905(1206010)) minus 119866

ℎ(120576119905))

+119908119905(119860119905(1206010) minus 119876119905) 119866ℎ(120576119905(1206010)) ]

+1

radic119899 minus 119906

119899

sum119905=119906+1

(119908119905minus 119908119905) 119860119905(1206010) 119866ℎ(120576119905(1206010))

= 1198701+ 1198702+ 1198703

(20)

For1198701 we have

1198701

radic119899 minus 119906=

1

119899 minus 119906

119899

sum119905=119906+1

119885119905119887119899119905

+ (1

119899 minus 119906

119899

sum119905=119906+1

119885119905)119874(ℎ

2)

(21)

where 119885119905= 119908119905119876119905 119887119899119905= 119866ℎ(120576119905) minus 119864(119866

ℎ(120576119905)) The second term

of (21) is 119874(radiclog 119899119899) as by the ergodicity Now turning tothe first term we suppose that 119876

119905is the first element 119880

119905minus1

without loss of generality Note that for each 119899 ge 119906 + 1119885119905119887119899119905F119905 119906 + 1 le 119905 le 119899 is a sequence of martingale

4 Journal of Applied Mathematics

differences with |119885119905119887119899119905| le 119862 where F

119905= 120590(120576

119904 119904 le 119905) For

some 1198620gt 0 by the ergodicity we have

1198812

119899=

119899

sum119905=119906+1

119864 (119885119905119887119899119905)2

| F119905minus1

=

119899

sum119905=119906+1

1198852

119905119864(119887119899119905)2

le 119862

119899

sum119905=119906+1

1198852

119905lt 119862 (119864 (119885

2

119905) + 1198620) 119899 as

(22)

Set119910 = 119862(119864(1198852119905)+1198620)119899 byTheorem 12A in [16] for all119860 gt 0

we have

119875

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum119905=119906+1

119885119905119887119899119905

1003816100381610038161003816100381610038161003816100381610038161003816

gt 119860radic119899 log 119899

= 119875

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum119905=119906+1

119885119905119887119899119905

1003816100381610038161003816100381610038161003816100381610038161003816

gt 119860radic119899 log 119899 1198812119899lt 119910 for some 119899

le 2 expminus1198602119899 log 119899

2 (119910 + 119862119860radic119899 log 119899)

= 2 expminus1198602 log 119899

2119862 (119864 (1198852119905) + 1198620) + 2119862119860radiclog 119899119899

(23)

Choosing 119860 such that 1198602 gt 2119862(119864(1198852119905) + 1198620) by the Borel-

Cantelli lemma the first term of (21) is119874(radiclog 119899119899) asThus1198701is 119874(radiclog 119899) as For 119870

2 by Davis [2] it holds that |120576

119905minus

120576119905(1206010)| le 120585119905 and 119860

119905(1206010)minus119876119905 le 120585119905 where 120585

119905= 119862sum

infin

119895=119905119903119895|119910119905minus119895|

for some 0 lt 119903 lt 1 Therefore

100381710038171003817100381711987021003817100381710038171003817 le

119862

ℎradic119899 minus 119906

119899

sum119905=119906+1

119908119905

100381710038171003817100381711987611990510038171003817100381710038171003816100381610038161003816120576119905 minus 120576119905 (1206010)

1003816100381610038161003816

+119862

radic119899 minus 119906

119899

sum119905=119906+1

119908119905

1003817100381710038171003817119876119905 minus 119860 119905 (1206010)1003817100381710038171003817

le119862

ℎradic119899 minus 119906

119899

sum119905=119906+1

120585119905+

119862

radic119899 minus 119906

119899

sum119905=119906+1

120585119905

=119862

ℎradic119899 minus 119906

119899

sum119905=119906+1

119903119905

infin

sum119897=0

119903119897 1003816100381610038161003816119910minus119897

1003816100381610038161003816

+119862

radic119899 minus 119906

119899

sum119905=119906+1

119903119905

infin

sum119897=0

119903119897 1003816100381610038161003816119910minus119897

1003816100381610038161003816as997888rarr 0

(24)

Thus1198702is 119874(radiclog 119899) as For119870

3 we have

100381710038171003817100381711987031003817100381710038171003817

le119862

radic119899 minus 119906

119899

sum119905=119906+1

1003816100381610038161003816119908119905 minus 11990811990510038161003816100381610038161003817100381710038171003817119860 119905 (1206010)

1003817100381710038171003817

le119862

radic119899 minus 119906

119899

sum119905=119906+1

suminfin

119896=119905119896minus120572

1003816100381610038161003816119910119905minus1198961003816100381610038161003816 sum119905minus1

119895=111990311989510038161003816100381610038161003816119910119905minus119895

10038161003816100381610038161003816

1 + sum119905minus1

119896=1119896minus120572

1003816100381610038161003816119910119905minus1198961003816100381610038161003816

le119862

radic119899 minus 119906

119899

sum119905=119906+1

infin

sum119896=119905

119896minus120572 1003816100381610038161003816119910119905minus119896

1003816100381610038161003816

119905minus1

sum119895=1

119903119895119895120572

le119862

radic119899 minus 119906

119899

sum119905=119906+1

infin

sum119896=119905

119896minus120572 1003816100381610038161003816119910119905minus119896

1003816100381610038161003816

le119862

radic119899 minus 119906

119899

sum119905=119906+1

infin

sum119897=0

(119905 + 119897)minus120572 1003816100381610038161003816119910minus119897

1003816100381610038161003816

le119862

radic119899 minus 119906

119899

sum119905=119906+1

(119905minus120572 10038161003816100381610038161199100

1003816100381610038161003816 +

infin

sum119897=1

2minus120572119905minus1205722

119897minus1205722 1003816100381610038161003816119910minus119897

1003816100381610038161003816)

=119862

radic119899 minus 119906

119899

sum119905=119906+1

119905minus120572 10038161003816100381610038161199100

1003816100381610038161003816

+119862

radic119899 minus 119906

119899

sum119905=119906+1

119905minus1205722

infin

sum119897=1

119897minus1205722 1003816100381610038161003816119910minus119897

1003816100381610038161003816as997888rarr 0

(25)

because we have the facts that 119860119905(1206010) le 119862sum

119905minus1

119895=1119903119895|119910119905minus119895| (see

[2]) and (119905+ 119897)minus120572 le 2minus120572(119905119897)minus1205722 for 119905 gt 0 119897 gt 0 Thus119870

3is also

119874(radiclog 119899) as Therefore part (i) holds For the proof of part(ii) we may write

120597119876119899ℎ(1206010)

120597120601120591

= minus1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119908119905119860119905(1206010) 119860120591

1119905(1206010)

times [119870(120576119905(1206010)

ℎ) + 119870(minus

120576119905(1206010)

ℎ)]

+1

119899 minus 119906

119899

sum119905=119906+1

119908119905

120597119860119905(1206010)

120597120601120591119866ℎ(120576119905(1206010))

= 1198631+ 1198632

(26)

where 1198601119905(120601) = minus120597120576

119905(120601)120597120601 For119863

1 we may write

1198631= minus

1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119908119905119876119905119876120591

1119905(119870(

120576119905

ℎ) + 119870(minus

120576119905

ℎ))

minus1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

[119908119905119860119905(1206010) 119860120591

1119905(1206010)

times (119870(120576119905(1206010)

ℎ) + 119870(minus

120576119905(1206010)

ℎ))

minus119908119905119876119905119876120591

1119905(119870(

120576119905

ℎ) + 119870(minus

120576119905

ℎ))]

= 11986311+ 11986312

(27)

Journal of Applied Mathematics 5

Note that

11986311= minus

1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119879119905119888119899119905minus

1

(119899 minus 119906) ℎ

times

119899

sum119905=119906+1

119908119905119876119905119876120591

1119905119864(119870(

120576119905

ℎ) + 119870(minus

120576119905

ℎ))

= minus1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119879119905119888119899119905

minus [1

119899 minus 119906

119899

sum119905=119906+1

119908119905119876119905119876120591

1119905] (2119891 (0) + 119900 (ℎ))

(28)

where 119879119905= 1199081199051198761199051198761205911119905and 119888119899119905= 119870(120576

119905ℎ) + 119870(minus120576

119905ℎ) minus 119864[119870(120576

119905

ℎ) + 119870(minus120576119905ℎ)] The second term of (28) is minus2119891(0)Σ

1as by

the ergodicity We will prove that the first term is 119900(1) asWe suppose that 119876

119905is the first element 119880

119905minus1without loss of

generality Note that for each 119899 ge 119906 + 1 119879119905119888119899119905F119905 119906 + 1 le

119905 le 119899 is a sequence ofmartingale differences with |119879119905119888119899119905| le 119862

and

1198812

119899=

119899

sum119905=119906+1

119864 (119879119905119888119899119905)2

| F119905minus1

=

119899

sum119905=119906+1

1198792

119905119864(119888119899119905)2

lt 119899119862 (119891 (0) 1198620ℎ + 119874 (ℎ

2)) as

(29)

where 1198620gt 0 is a constant Set 119910 = 119899119862(119891(0)119862

0ℎ + 119874(ℎ2)) by

Theorem 12A in [16] for all 120576 gt 0 we have

119875

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum119905=119906+1

119879119905119888119899119905

1003816100381610038161003816100381610038161003816100381610038161003816

gt (119899ℎ) 120576

= 119875

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum119905=119906+1

119879119905119888119899119905

1003816100381610038161003816100381610038161003816100381610038161003816

gt (119899ℎ) 120576 1198812

119899lt 119910 for some 119899

le 2 exp minus(119899ℎ)21205762

2 (119910 + 119862119899ℎ120576)

= 2 exp minus119899ℎ1205762

2119862 (119891 (0) 1198620+ 119874 (ℎ)) + 2119862120576

(30)

The result follows from the Borel-Cantelli lemmaThus11986311=

minus2119891(0)Σ1+ 119900(1) as Similar to119870

2and119870

3 we have

1003817100381710038171003817119863121003817100381710038171003817

le1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119908119905(1003817100381710038171003817119876119905 minus 119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198761119905

1003817100381710038171003817

+1003817100381710038171003817119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198761119905 minus 1198601119905 (1206010)

1003817100381710038171003817)

100381610038161003816100381610038161003816100381610038161003816119870(

120576119905(1206010)

ℎ) + 119870(minus

120576119905(1206010)

ℎ)

100381610038161003816100381610038161003816100381610038161003816

+1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

1003816100381610038161003816119908119905 minus 11990811990510038161003816100381610038161003817100381710038171003817119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198601119905 (1206010)

1003817100381710038171003817

times

100381610038161003816100381610038161003816100381610038161003816119870(

120576119905(1206010)

ℎ) + 119870(minus

120576119905(1206010)

ℎ)

100381610038161003816100381610038161003816100381610038161003816

+1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119908119905

1003817100381710038171003817119876119905100381710038171003817100381710038171003817100381710038171198761119905

1003817100381710038171003817

times

100381610038161003816100381610038161003816100381610038161003816[119870(

120576119905

ℎ) + 119870(minus

120576119905

ℎ)]

minus[119870(120576119905(1206010)

ℎ) + 119870(minus

120576119905(1206010)

ℎ)]

100381610038161003816100381610038161003816100381610038161003816

le119862

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119908119905(1003817100381710038171003817119876119905 minus 119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198761119905

1003817100381710038171003817

+1003817100381710038171003817119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198761119905 minus 1198601119905 (1206010)

1003817100381710038171003817)

+119862

(119899 minus 119906) ℎ

119899

sum119905=119906+1

1003816100381610038161003816119908119905 minus 11990811990510038161003816100381610038161003817100381710038171003817119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198601119905 (1206010)

1003817100381710038171003817

+119862

(119899 minus 119906) ℎ2

119899

sum119905=119906+1

119908119905

1003817100381710038171003817119876119905100381710038171003817100381710038171003817100381710038171198761119905

10038171003817100381710038171003816100381610038161003816120576119905 minus 120576119905 (1206010)

1003816100381610038161003816

le119862

(119899 minus 119906) ℎ

119899

sum119905=119906+1

120585119905+

119862

(119899 minus 119906) ℎ

119899

sum119905=119906+1

infin

sum119896=119905

119896minus120572 1003816100381610038161003816119910119905minus119896

1003816100381610038161003816

+119862

(119899 minus 119906) ℎ2

119899

sum119905=119906+1

120585119905

as997888rarr 0

(31)

Therefore 1198631

= minus2119891(0)Σ1+ 119900(1) as For 119863

2 from the

definition of 119860119905(120601) it holds for 119905 gt max(119901 119902) that

120579 (119861)120597119860119905119894(120601)

120597120601119895

= 0 119894 119895 = 1 119901

120579 (119861)120597119860119905119894(120601)

120597120601119895+119901

= minus119860119905minus119895119894

(120601) 119894 = 1 119901 119895 = 1 119902

120579 (119861)120597119860119905119895+119901

(120601)

120597120601119894

= minus119860119905minus119895119894

(120601) 119894 = 1 119901 119895 = 1 119902

120579 (119861)120597119860119905119894+119901

(120601)

120597120601119895+119901

= minus119860119905minus119895119894+119901

(120601) minus 119860119905minus119894119895+119901

(120601)

119894 119895 = 1 119902

(32)

6 Journal of Applied Mathematics

where 119861 is the backshift operator For 119905 = 0 plusmn1 plusmn2 define

1205790(119861)119883119905(119894119895)

= 0 119894 119895 = 1 119901

1205790(119861)119883119905(119894119895+119901)

= minus119876119905minus119895119894

119894 = 1 119901 119895 = 1 119902

1205790(119861)119883119905(119895+119901119894)

= minus119876119905minus119895119894

119894 = 1 119901 119895 = 1 119902

1205790(119861)119883119905(119894+119901119895+119901)

= minus119876119905minus119895119894+119901

minus 119876119905minus119894119895+119901

119894 119895 = 1 119902

(33)

where 119876119905119894

is the 119894th component of 119876119905 Put 119883

119905= (119883

119905(119894119895))

similar to [13] we have that 120597119860119905(1206010)120597120601120591 le 119862sum

119905minus1

119895=1119903119895|119910119905minus119895|

119883119905 le 119862sum

infin

119895=1119903119895|119910119905minus119895| and 119883

119905minus 120597119860119905(1206010)120597120601120591 le 120585

119905 Then

we may write

1198632=

1

119899 minus 119906

119899

sum119905=119906+1

119908119905119883119905119866ℎ(120576119905)

+1

119899 minus 119906

119899

sum119905=119906+1

[119908119905

120597119860119905(1206010)

120597120601120591119866ℎ(120576119905(1206010)) minus 119908

119905119883119905119866ℎ(120576119905)]

= 11986321+ 11986322

(34)

Similar to11986311and119863

12 we have that119863

21and119863

22are 119900(1) as

This completes the proof

Lemma 4 Under the conditions of Proposition 1 as 119899 rarr infin

(119894) 119885119899(120601) = 119900 (119899

13) as

(119894119894) 119878 (120601) = Ω + 119900 (1) as(35)

hold uniformly in 119861

Proof of Lemma 4 For part (i) from [2] we havethat 120597119860

119905(120601)120597120601120591 le 119862sum

119905minus1

119895=1119903119895|119910119905minus119895| and 119860

119905(120601) le

119862sum119905minus1

119895=1119903119895|119910119905minus119895| uniformly hold in the ball 119861 for sufficiently

large 119899 Then for each 120601 isin 119861 we have1003817100381710038171003817119898119905ℎ (120601)

1003817100381710038171003817 le 119908119905

1003817100381710038171003817119860 119905 (120601)1003817100381710038171003817

le119862sum119905minus1

119895=111990311989510038161003816100381610038161003816119910119905minus119895

10038161003816100381610038161003816

1 + sum119905minus1

119896=1119896minus120572

1003816100381610038161003816119910119905minus1198961003816100381610038161003816

le 119862

119905minus1

sum119896=1

119903119896119896120572

le 119862

infin

sum119896=1

119903119896119896120572lt infin

(36)

Thus part (i) holds For part (ii) similar to the proof ofLemma 3 we have that 119878(120601

0) = Ω + 119900(1) as For each 120601 isin 119861

by Taylor expansion we have

119878 (120601) minus 119878 (1206010)

=1

119899 minus 119906

119899

sum119905=119906+1

119898119905ℎ(1206010)

120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)

120591

+1

119899 minus 119906

119899

sum119905=119906+1

120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)119898120591

119905ℎ(1206010)

+1

119899 minus 119906

119899

sum119905=119906+1

120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)

times 120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)

120591

= 1198791+ 1198792+ 1198793

(37)

where 120601lowast lies between 1206010and 120601 For 119879

1 we have

100381710038171003817100381711987911003817100381710038171003817 le

1

119899 minus 119906

119899

sum119905=119906+1

1199082

119905

1003817100381710038171003817119860 119905 (1206010)1003817100381710038171003817

100381710038171003817100381710038171003817100381710038171003817

120597119860119905(120601lowast)

120597120601120591

100381710038171003817100381710038171003817100381710038171003817

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

+1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

1199082

119905

1003817100381710038171003817119860 119905 (1206010)10038171003817100381710038171003817100381710038171003817119860 119905 (120601

lowast)1003817100381710038171003817

times10038171003817100381710038171198601119905 (120601

lowast)10038171003817100381710038171003817100381710038171003817120601 minus 1206010

1003817100381710038171003817

times

100381610038161003816100381610038161003816100381610038161003816119870(

120576119905(120601lowast)

ℎ) + 119870(minus

120576119905(120601lowast)

ℎ)

100381610038161003816100381610038161003816100381610038161003816

le119862119889119899

119899 minus 119906

119899

sum119905=119906+1

(

119905minus1

sum119895=1

119903119895119895120572)

2

+119862119889119899

(119899 minus 119906) ℎ

119899

sum119905=119906+1

(

119905minus1

sum119895=1

119903119895119895120572)

3

as997888rarr 0

(38)

Similarly we have 1198792

as997888997888rarr 0 and 119879

3

as997888997888rarr 0 This completes the

proof

Proof of Proposition 1 For 120601 isin 119861 by Taylor expansion

119876119899ℎ(120601) = 119876

119899ℎ(1206010) +

120597119876119899ℎ(1206010)

120597120601120591(120601 minus 120601

0)

+1

2

119901+119902minus119898

sum119895119896=1

1205972119876119899ℎ(1206010)

120597120601119895120597120601119896

(120601119895minus 1206010

119895) (120601119896minus 1206010

119896)

+1

6

119901+119902minus119898

sum119895119896119897=1

1205973119876119899ℎ(120601lowast)

120597120601119895120597120601119896120597120601119897

times (120601119895minus 1206010

119895) (120601119896minus 1206010

119896) (120601119897minus 1206010

119897)

(39)

where 120601lowastlies between 120601

0and 120601 Note that the final term on

the right side of (39) can be written as

1

6

119901+119902minus119898

sum119895119896119897=1

[

[

(120601119895minus 1206010119895) (120601119896minus 1206010119896) (120601119897minus 1206010119897)

ℎ3]

]

times (ℎ3

119899 minus 119906

119899

sum119905=119906+1

1205973119898119905ℎ(120601lowast)

120597120601119895120597120601119896120597120601119897

)

(40)

Journal of Applied Mathematics 7

which is 119900(120575119899) as where 120575

119899= 120601 minus 120601

0 because 1198892

119899ℎ3 =

11198992120573minus3120574 rarr 0 and1003817100381710038171003817100381710038171003817100381710038171003817

ℎ3

119899 minus 119906

119899

sum119905=119906+1

1205973119898119905ℎ(120601lowast)

120597120601119895120597120601119896120597120601119897

1003817100381710038171003817100381710038171003817100381710038171003817

le 119862 (41)

The third term on the right side of (39) can be written as

1

2

119901+119902minus119898

sum119895119896=1

[

[

(120601119895minus 1206010

119895) (120601119896minus 1206010

119896)

ℎ]

]

times ℎ

119899 minus 119906

119899

sum119905=119906+1

1205972119898119905ℎ(1206010)

120597120601119895120597120601119896

(42)

which is also 119900(120575119899) as because 119889

119899ℎ = 1119899120573minus120574 rarr 0 and

ℎ

119899 minus 119906

119899

sum119905=119906+1

1205972119898119905ℎ(1206010)

120597120601119895120597120601119896

= 119900 (1) as (43)

by a similar proof of Lemma 3 Therefore

119876119899ℎ(120601) = 119876

119899ℎ(1206010) +

120597119876119899ℎ(1206010)

120597120601120591(120601 minus 120601

0) + 119900 (120575

119899) as

(44)

uniformly about 120601 isin 119861 Denote 120601 = 1206010+ 120583119889119899 for 120601 isin 120601

120601 minus 1206010 = 119889

119899 where 120583 = 1 Now we give a lower bound

for 119897ℎ(120601) on the surface of the ball Similar to [6] by Lemmas

3 and 4 we have

119897ℎ(120601)

= (119899 minus 119906)119876119899ℎ(120601)120591

119878(120601)minus1

119876119899ℎ(120601) + 119900 (119899

43minus3120573) as

= (119899 minus 119906)[

[

119874(radiclog 119899119899

)

+ (minus2119891 (0)) Σ1120583119889119899+ 119900 (119889

119899) ]

]

120591

Ωminus1

times [

[

119874(radiclog 119899119899

) + (minus2119891 (0)) Σ1120583119889119899+ 119900 (119889

119899)]

]

+ 119900 (11989943minus3120573

) as

ge (119888 minus 120576) 1198991minus2120573 as

(45)

where 119888 minus 120576 gt 0 and 119888 is the smallest eigenvalue of41198912(0)Σ120591

1Ωminus1Σ1 Similarly

119897ℎ(1206010)

= (119899 minus 119906)119876119899ℎ(1206010)120591

119878(1206010)minus1

119876119899ℎ(1206010) + 119900 (1) as

= 119874 (log 119899) as

(46)

Since 119897ℎ(120601) is a continuous function about 120601 as 120601 belongs to

the ball 119861 119897ℎ(120601) attains its minimum value at some point 120601

in the interior of this ball and 120601 satisfies 120597119897ℎ(120601)120597120601 = 0 it

follows that (12) holds This completes the proof

Proof of Theorem 2 Similar to the proof ofTheorem 2 of Qinand Lawless [17] we have

(

120601 minus 1206010

) = 119878minus1

119899(minus1198761119899(1206010 0) + 119900

119901(119899minus12)

119900119901(119899minus12

)) (47)

where

119878119899= (

1205971198761119899(1206010 0)

120597120582120591

1205971198761119899(1206010 0)

120597120601120591

1205971198762119899(1206010 0)

1205971205821205910

)119901

997888rarr (11987811

11987812

11987821

0)

= (minusΩ minus2119891 (0) Σ

1

minus2119891 (0) Σ120591

10

)

(48)

By the standard arguments in the proof of empirical likeli-hood (see [6]) we have

119897119901(1205960)

= minus (119899 minus 119906) (119876119899ℎ(1206010)120591

0) 119878minus1

119899(119876119899ℎ(1206010)120591

0)120591

+ 119900119901(1)

= (Ωminus12radic119899 minus 119906119876

119899ℎ(1206010))120591

times (119868 minus 41198912(0)Ωminus12

Σ1Δminus1Σ120591

1Ωminus12

)

times (Ωminus12radic119899 minus 119906119876

119899ℎ(1206010)) + 119900

119901(1)

(49)

where Δ = 41198912(0)Σ1205911Ωminus1Σ1 Sinceradic119899 minus 119906119876

119899ℎ(1206010)119889

997888rarr 119873(0Ω)

and

tr 41198912 (0)Ωminus12Σ1Δminus1Σ120591

1Ωminus12

= tr Δminus141198912 (0) Σ1205911Ωminus1Σ1 = 119901 + 119902 minus 119898

(50)

it follows that 119897119901(1205960)119889

997888rarr 1205942119898

4 Simulation Studies

We generated data from a simple ARMA(1 1) model 119910119905=

1205931119910119905minus1

+ 120576119905+ 1205991120576119905minus1

with119873(0 1) 1199052 and Cauchy innovation

distribution We set 119906 = 20 120572 = 3 and the true value(1205931 1205991) = (04 07) or (minus05 07) where 120593

1is the parameter

of interest The sample size 119899 = 50 100 150 200 and2000 replications are conducted in all cases We smooth theestimating equations using kernel

119870 (119909) =1

radic2120587120590119890minus119909221205902

(51)

8 Journal of Applied Mathematics

Table 1 The coverage probability of confidence intervals when 120576119905sim 119873(0 1)

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08818 08820 08822 08193 07875100 08898 08896 08897 08655 08431150 08926 08927 08932 08692 08395200 08983 08983 08986 08666 08363

119886 = 09

1205931= minus05

50 08888 08885 08892 08156 07813100 08967 08968 08973 08738 08448150 08943 08944 08946 08823 08574200 08972 08979 08977 08931 08692

119886 = 095

1205931= 04

50 09347 09350 09350 08724 08425100 09424 09430 09431 09123 08862150 09467 09471 09470 09157 08936200 09494 09494 09497 09160 08937

119886 = 095

1205931= minus05

50 09404 09404 09404 08705 08407100 09472 09474 09474 09134 08931150 09481 09479 09476 09248 09052200 09495 09495 09490 09326 09152

where 120590 = 01 which is the so-called Gaussian kernel Thecoverage probabilities of smoothed empirical likelihood con-fidence regions 119868

ℎ119888with the bandwidth ℎ = 1119899120574 are denot-

ed by EL(120574) where 120574 = 027 030 032 respectivelyAs another benchmark of the simulation experiments

we consider the confidence regions based on the asymptoticnormal distribution ofWLADEproposed by [3] To constructthe confidence regions we need to estimate 119891(0) Σ and ΩWe can estimate 119891(0) by

119891 (0) =1

119908119887119899(119899 minus 119906)

119899

sum119905=119906+1

119908119905

120576119905(120579)

119887119899

(52)

where (119909) = exp(minus119909)(1+exp(minus119909))2 is a kernel function on119877 and 119887

119899= 1119899] is a bandwidth

119908= (119899 minus 119906)

minus1sum119899

119905=119906+1119908119905 Σ

andΩ can be estimated respectively by

Σ =1

119899 minus 119906

119899

sum119905=119906+1

119908119905119876119905119876120591

119905

Ω =1

119899 minus 119906

119899

sum119905=119906+1

1199082

119905119876119905119876120591

119905

(53)

where 119876119905is defined in the same manner as 119876

119905 1205790is replaced

by 120579 and 120576119905is replaced by 120576

119905(120579) see (14) Based on this we can

construct a NA confidence region (ie based on the normalapproximation of WLADE) The coverage probabilities ofconfidence regions 119868NA based on the bandwidth 119887

119899= 1119899] are

denoted byNA(]) with ] = 025 020 respectively Tables 12

and 3 show the probabilities of the confidence intervals of 1205931

at confidence levels 09 and 095 respectivelyThe simulation results can be summarized as followsThe

coverage probabilities of NA(]) are much smaller than thenominal levels and very sensitive to the choice of bandwidth119887119899and 120576119905 On the other hand the coverage probabilities of

EL(120574) are much better and less sensitive to the choice ofbandwidth ℎ and 120576

119905 As the sample size 119899 increases the

coverage probabilities for both increase to the nominal levelsas one might expect

5 Conclusions

This paper explores a profile empirical likelihood method toconstruct confidence regions for the partial parameters ofinterest in IVARMAmodels We started with the foundationof estimating equations of WLADE then from there wederived smoothed empirical likelihood Moreover we haveproved that the resulting statistics has asymptotic standardchi-squared distribution Hence there is no need to estimateany additional quantity such as the asymptotic variance Thesimulations indeed show that the proposed method has agood finite sample behavior which experimentally confirmsour method

Conflict of Interests

The authors declare that they have no conflict of interests

Journal of Applied Mathematics 9

Table 2 The coverage probability of confidence intervals when 120576119905sim 1199052

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08774 08776 08779 07627 07290100 08850 08849 08856 08323 07995150 08919 08917 08924 08467 08157200 08935 08932 08932 08510 08229

119886 = 09

1205931= minus05

50 08902 08902 08899 07295 07055100 08942 08944 08950 08151 07836150 08946 08941 08937 08473 08204200 08965 08963 08961 08641 08392

119886 = 095

1205931= 04

50 09331 09327 09327 08163 07854100 09412 09412 09413 08834 08542150 09447 09446 09447 08965 08718200 09456 09458 09459 09001 08762

119886 = 095

1205931= minus05

50 09421 09418 09418 07963 07658100 09455 09462 09460 08695 08430150 09444 09442 09440 08910 08695200 09464 09463 09464 09072 08863

Table 3 The coverage probability of confidence intervals when 120576119905sim 119862119886119906119888ℎ119910

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08360 08358 08361 06345 06018100 08708 08708 08702 07286 06942150 08811 08819 08823 07614 07270200 08870 08864 08868 07800 07467

119886 = 09

1205931= minus05

50 08673 08679 08678 05818 05502100 08869 08870 08874 06756 06440150 08972 08974 08974 07232 06867200 08956 08955 08953 07510 07156

119886 = 095

1205931= 04

50 08992 08994 08996 06888 06557100 09263 09261 09263 07870 07530150 09362 09366 09367 08176 07857200 09402 09398 09399 08350 08044

119886 = 095

1205931= minus05

50 09235 09239 09240 06378 06065100 09422 09420 09420 07342 07030150 09474 09479 09481 07820 07530200 09467 09471 09470 08057 07778

Acknowledgments

Lirsquos research is partially supported by the FundamentalResearch Funds for the Central Universities (no 2013XK03)Hersquos work is partially supported by the National NaturalScience Foundation of China (nos 11171230 and 11231010)

References

[1] TMikosch TGadrich C Kluppelberg andR J Adler ldquoParam-eter estimation for ARMA models with infinite variance inno-vationsrdquoTheAnnals of Statistics vol 23 no 1 pp 305ndash326 1995

[2] R A Davis ldquoGauss-Newton and M-estimation for ARMAprocesses with infinite variancerdquo Stochastic Processes and theirApplications vol 63 no 1 pp 75ndash95 1996

[3] J Pan HWang andQ Yao ldquoWeighted least absolute deviationsestimation forARMAmodels with infinite variancerdquoEconomet-ric Theory vol 23 no 5 pp 852ndash879 2007

[4] K Zhu and S Ling ldquoThe global LAD estimators for finiteinfinite variance ARMA(pq) modelsrdquo Econometric Theory vol28 no 5 pp 1065ndash1086 2012

[5] A B Owen ldquoEmpirical likelihood ratio confidence intervals fora single functionalrdquo Biometrika vol 75 no 2 pp 237ndash249 1988

[6] A BOwen ldquoEmpirical likelihood ratio confidence regionsrdquoTheAnnals of Statistics vol 18 no 1 pp 90ndash120 1990

10 Journal of Applied Mathematics

[7] A B Owen Empirical Likelihood Chapman and Hall LondonUK 2001

[8] A C Monti ldquoEmpirical likelihood confidence regions in timeseries modelsrdquo Biometrika vol 84 no 2 pp 395ndash405 1997

[9] C Chuang and N H Chan ldquoEmpirical likelihood for autore-gressive models with applications to unstable time seriesrdquoStatistica Sinica vol 12 no 2 pp 387ndash407 2002

[10] N H Chan L Peng and Y Qi ldquoQuantile inference for near-integrated autoregressive time series with infinite variancerdquoStatistica Sinica vol 16 no 1 pp 15ndash28 2006

[11] J Li W Liang S He and X Wu ldquoEmpirical likelihood forthe smoothed LAD estimator in infinite variance autoregressivemodelsrdquo Statistics amp Probability Letters vol 80 no 17-18 pp1420ndash1430 2010

[12] J Li W Liang and S He ldquoEmpirical likelihood for LADestimators in infinite variance ARMA modelsrdquo Statistics ampProbability Letters vol 81 no 2 pp 212ndash219 2011

[13] P J Brockwell and R A DavisTime seriesTheory andMethodsSpringer-Verlag New York NY USA 2nd edition 1991

[14] P C B Phillips ldquoA shortcut to LAD estimator asymptoticsrdquoEconometric Theory vol 7 no 4 pp 450ndash463 1991

[15] B W Silverman Density Estimation for Statistics and DataAnalysis Chapman ampHall London UK 1986

[16] V H de la Pena ldquoA general class of exponential inequalities formartingales and ratiosrdquoThe Annals of Probability vol 27 no 1pp 537ndash564 1999

[17] J Qin and J Lawless ldquoEmpirical likelihood and general estimat-ing equationsrdquo The Annals of Statistics vol 22 no 1 pp 300ndash325 1994

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Stochastic AnalysisInternational Journal of

Page 2: Research Article Empirical Likelihood for Partial ...downloads.hindawi.com/journals/jam/2014/868970.pdf · Research Article Empirical Likelihood for Partial Parameters in ARMA Models

2 Journal of Applied Mathematics

likelihood to near unit root AR(1) model with infinite vari-ance errors Li et al [11 12] respectively used the empiricallikelihood to infinite variance AR(119901) models and model (1)

The rest of the paper is organized as follows In Section 2we propose the profile empirical likelihood for the parametersof interest and show the main result Section 3 provides theproofs of the main results Some simulations are conductedin Section 4 to illustrate our approach Conclusions are givenin Section 5

2 Methodology and Main Results

First the parameter space is denoted by Θ sub 119877119901+119902 whichcontains the true value 120579

0of the parameter 120579 as an inner point

For 120579 = (1205931 120593

119901 1205991 120599

119902) put

120576119905(120579) =

119910119905minus

119901

sum119894=1

120593119894119910119905minus119894

minus

119902

sum119895=1

120599119895120576119905minus119895

(120579) if 119905 gt 0

0 if 119905 le 0

(2)

where 119910119905equiv 0 for all 119905 le 0 and note that 120576

119905(1205790) = 120576119905 because

of this truncationWe define the objective function as

119878119899(120579) =

119899

sum119905=119906+1

119908119905

1003816100381610038161003816120576119905 (120579)1003816100381610038161003816 (3)

where 119906 ge max(119901 119902) and the weight function 119908119905= 1(1+

sum119905minus1

119896=1119896minus120572|119910119905minus119896|)4 depending on a constant 120572 gt 2 The

WLADE denoted by 120579 is a lacol minimizer of 119878119899(120579) in

a neighborhood of 1205790[3] Denote 119860

119905(120579) = (119860

1199051(120579)

119860119905119901+119902

(120579))120591 where119860

119905119894(120579) = minus120597120576

119905(120579)120597120579

119894 By (8119) of Brock-

well and Davis [13] it holds for 119905 gt max(119901 119902) that

119860119905119894(120579) +

119902

sum119895=1

120599119895119860119905minus119895119894

(120579) = 119910119905minus119894 119894 = 1 119901

119860119905119894+119901

(120579) +

119902

sum119895=1

120599119895119860119905minus119895119894+119901

(120579) = 120576119905minus119894(120579) 119894 = 1 119902

(4)

Hence 120579 satisfies estimating equation

1

119899 minus 119906

119899

sum119905=119906+1

119908119905119860119905(120579) sgn 120576

119905(120579) = 0 (5)

where sgn(119909) = minus1 for 119909 lt 0 and =1 for 119909 ge 0

(see [14]) Note that the above estimating equation is notdifferentiable at point 120579 such that 120576

119905(120579) = 0 for some 119905

This causes some problems for our subsequent asymptoticanalysis To overcome this problem we replace it with asmooth function Define a probability density kernel119870(sdot) [15]such that int+infin

minusinfin119909119895119870(119909)119889119909 = 0 120581 for 119895 = 1 2 respectively

where 120581 = 0 Let 119866ℎ(119909) = int

119909ℎ

minus119909ℎ119870(119906)119889119906 for ℎ gt 0 Then a

smoothed version of (5) is

1

119899 minus 119906

119899

sum119905=119906+1

119908119905119860119905(120579) 119866ℎ(120576119905(120579)) = 0 (6)

Let 119898119905ℎ(120579) = 119908

119905119860119905(120579)119866ℎ(120576119905(120579)) a smoothed empirical

log-likelihood ratio is defined as

119897ℎ(120579)

= minus2 sup119899

sum119905=119906+1

log ((119899 minus 119906) 119901119905) |

119899

sum119905=119906+1

119901119905119898119905ℎ(120579) = 0

119901119905ge 0

119899

sum119905=119906+1

119901119905= 1

(7)

Using the Lagrange multiplier the optimal value of 119901119905is

derived to be

119901119905(120579) =

1

(119899 minus 119906) (1 + 120582(120579)120591119898119905ℎ(120579))

119906 + 1 le 119905 le 119899

(8)

where 120582(120579) is a 119901 + 119902-dimensional vector of Lagrangemultipliers satisfying

1

119899 minus 119906

119899

sum119905=119906+1

119898119905ℎ(120579)

1 + 120582(120579)120591119898119905ℎ(120579)

= 0 (9)

This gives the smoothed empirical log-likelihood ratio statis-tic

119897ℎ(120579) = 2

119899

sum119905=119906+1

log (1 + 120582(120579)120591119898119905ℎ(120579)) (10)

Let 120579 = (120601120591 120596120591)120591 where 120596 isin 119877119898 (1 le 119898 le 119901 + 119902)

is the parameter of interest and 120601 isin 119877119901+119902minus119898 is the nuisanceparameter Note that 119898 = 119901 + 119902 means no nuisance param-eters Let 120601

0and 120596

0denote the true values of 120601 and 120596 respec-

tively The profile empirical likelihood is defined as

119897119901(120596) = min

120601

119897ℎ(120601 120596) (11)

That is 119897119901(120596) = 119897

ℎ(120601(120596) 120596) where 120601 = 120601(120596) = argmin

120601119897ℎ

(120601 120596)The following conditions are in order

(A1) The characteristic polynomial 120601(119911) = 1 minus 1205931119911 minus sdot sdot sdot minus

120593119901119911119901 and 120579(119911) = 1+120599

1119911+ sdot sdot sdot + 120599

119902119911119902 have no common

zeros and all roots of 120601(119911) and 120579(119911) are outside theunit circle

(A2) The innovation 120576119905 has zero median and a differen-

tiable density 119891(119909) satisfying the conditions 119891(0) gt 0sup119909isin119877

|119891(119909)| lt 1198611lt infin and sup

119909isin119877|1198911015840(119909)| lt 119861

2lt

infin Furthermore 119864|120576119905|120575 lt infin for some 120575 gt 0 and

120572 gt max2 2120575(A3) As 119899 rarr infin 119906 rarr infin and 119906119899 rarr 0(A4) The second derivative of 119870 exists in 119877 and 1198701015840(119909) and

11987010158401015840(119909) are bounded(A5) ℎ = 1119899120574 with 14 lt 120574 lt 13

First we show the existence and consistency of 120601(1205960)

Journal of Applied Mathematics 3

Proposition 1 Let 119889119899= 1119899120573 with max13 31205742 lt 120573 lt

12 Assume (A1)ndash(A5) hold then as 119899 rarr infin with probability1 there exists a local minimizer 120601 of 119897

ℎ(120601 1205960) which lies in the

interior of the ball 119861 = 120601 120601 minus 1206010 le 119889

119899 Moreover 120601 and

= 120582(120601 1205960) satisfy

1198761119899(120601 ) = 0 119876

2119899(120601 ) = 0 (12)

where

1198761119899(120601 120582) =

1

119899 minus 119906

119899

sum119905=119906+1

119898119905ℎ(120601 1205960)

1 + 120582120591119898119905ℎ(120601 1205960)

1198762119899(120601 120582)

=1

119899 minus 119906

119899

sum119905=119906+1

1

1 + 120582120591119898119905ℎ(120601 1205960)(120597119898119905ℎ(120601 1205960)

120597120601120591)

120591

120582

(13)

The following theorem presents the asymptotic distribu-tion of the profile empirical likelihood

Theorem 2 Under conditions of Proposition 1 as 119899 rarr infinthe random variable 119897

119901(1205960) with 120601 given in Proposition 1

converges in distribution to 1205942119898

If 119888 is chosen such that 119875(1205942119898le 119888) = 119886 then Theorem 2

implies that the asymptotic coverage probability of empiricallikelihood confidence region 119868

ℎ119888= (120596 119897

119901(120596) le 119888) will be 119886

that is 119875(1205960isin 119868ℎ119888) = 119875(119897

119901(1205960) le 119888) = 119886 + 119900(1) as 119899 rarr infin

3 Proofs of the Main Results

In the following sdot denotes the Euclidian norm for a vectoror matrix and 119862 denotes a positive constant which may bedifferent at different places For 119905 = 0 plusmn1 plusmn2 define

119880119905minus

119901

sum119894=1

1205930

119894119880119905minus119894

= 120576119905 119881

119905+

119902

sum119895=1

1205990

119895119881119905minus119895

= 120576119905 (14)

Put 119876119905

= (119880119905minus1 119880

119905minus119901 119881119905minus1 119881

119905minus119902)120591 119908119905

= 1(1 +

suminfin

119896=1119896minus120572|119910119905minus119896|)4 and the corresponding partial vector for 120601

0

is denoted by 1198761119905 Let

Σ = 119864 (119908119905119876119905119876120591

119905)

Σ1= 119864 (119908

119905119876119905119876120591

1119905)

Ω = 119864 (1199082

119905119876119905119876120591

119905)

(15)

Assumptions A1 and A2 imply that for 120575 = min(120575 1)

119864(

infin

sum119896=1

119896minus1205722 1003816100381610038161003816119910119905minus119896

1003816100381610038161003816)

120575

le

infin

sum119896=1

119896minus1205721205752

1198641003816100381610038161003816119910119905minus119896

1003816100381610038161003816120575

lt infin (16)

Hence suminfin119896=1

119896minus1205722|119910119905minus119896| lt infin with probability 1 which

ensures that 119908119905

is well defined Note that 119876119905 le

119862suminfin

119895=1119903119895|119910119905minus119895| for some 0 lt 119903 lt 1 and

11990812

119905

10038171003817100381710038171198761199051003817100381710038171003817 le

119862suminfin

119895=111990311989510038161003816100381610038161003816119910119905minus119895

10038161003816100381610038161003816

1 + suminfin

119896=1119896minus120572

1003816100381610038161003816119910119905minus1198961003816100381610038161003816le 119862

infin

sum119896=1

119903119896119896120572lt infin (17)

Then Σ Σ1 and Ω are well-defined (finite) matrices For

simplicity we denote (120601 1205960) and (120601

0 1205960) by 120601 and 120601

0

respectively in this section The following notations will beused in the proofs Let

119885119899(120601) = max

119906+1le119905le119899

1003817100381710038171003817119898119905ℎ (120601)1003817100381710038171003817

119876119899ℎ(120601) =

1

119899 minus 119906

119899

sum119905=119906+1

119898119905ℎ(120601)

119878 (120601) =1

119899 minus 119906

119899

sum119905=119906+1

119898119905ℎ(120601)119898

119905ℎ(120601)120591

(18)

To prove Proposition 1 we first prove the following lemmas

Lemma 3 Under the conditions of Proposition 1 as 119899 rarr infin

(119894) 119876119899ℎ(1206010) = 119874(radic

log 119899119899

) as

(119894119894)120597119876119899ℎ(1206010)

120597120601120591= minus2119891 (0) Σ

1+ 119900 (1) as

(19)

Proof of Lemma 3 For part (i) we may write

radic119899 minus 119906119876119899ℎ(1206010)

=1

radic119899 minus 119906

119899

sum119905=119906+1

119908119905119876119905119866ℎ(120576119905)

+1

radic119899 minus 119906

119899

sum119905=119906+1

[119908119905119876119905(119866ℎ(120576119905(1206010)) minus 119866

ℎ(120576119905))

+119908119905(119860119905(1206010) minus 119876119905) 119866ℎ(120576119905(1206010)) ]

+1

radic119899 minus 119906

119899

sum119905=119906+1

(119908119905minus 119908119905) 119860119905(1206010) 119866ℎ(120576119905(1206010))

= 1198701+ 1198702+ 1198703

(20)

For1198701 we have

1198701

radic119899 minus 119906=

1

119899 minus 119906

119899

sum119905=119906+1

119885119905119887119899119905

+ (1

119899 minus 119906

119899

sum119905=119906+1

119885119905)119874(ℎ

2)

(21)

where 119885119905= 119908119905119876119905 119887119899119905= 119866ℎ(120576119905) minus 119864(119866

ℎ(120576119905)) The second term

of (21) is 119874(radiclog 119899119899) as by the ergodicity Now turning tothe first term we suppose that 119876

119905is the first element 119880

119905minus1

without loss of generality Note that for each 119899 ge 119906 + 1119885119905119887119899119905F119905 119906 + 1 le 119905 le 119899 is a sequence of martingale

4 Journal of Applied Mathematics

differences with |119885119905119887119899119905| le 119862 where F

119905= 120590(120576

119904 119904 le 119905) For

some 1198620gt 0 by the ergodicity we have

1198812

119899=

119899

sum119905=119906+1

119864 (119885119905119887119899119905)2

| F119905minus1

=

119899

sum119905=119906+1

1198852

119905119864(119887119899119905)2

le 119862

119899

sum119905=119906+1

1198852

119905lt 119862 (119864 (119885

2

119905) + 1198620) 119899 as

(22)

Set119910 = 119862(119864(1198852119905)+1198620)119899 byTheorem 12A in [16] for all119860 gt 0

we have

119875

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum119905=119906+1

119885119905119887119899119905

1003816100381610038161003816100381610038161003816100381610038161003816

gt 119860radic119899 log 119899

= 119875

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum119905=119906+1

119885119905119887119899119905

1003816100381610038161003816100381610038161003816100381610038161003816

gt 119860radic119899 log 119899 1198812119899lt 119910 for some 119899

le 2 expminus1198602119899 log 119899

2 (119910 + 119862119860radic119899 log 119899)

= 2 expminus1198602 log 119899

2119862 (119864 (1198852119905) + 1198620) + 2119862119860radiclog 119899119899

(23)

Choosing 119860 such that 1198602 gt 2119862(119864(1198852119905) + 1198620) by the Borel-

Cantelli lemma the first term of (21) is119874(radiclog 119899119899) asThus1198701is 119874(radiclog 119899) as For 119870

2 by Davis [2] it holds that |120576

119905minus

120576119905(1206010)| le 120585119905 and 119860

119905(1206010)minus119876119905 le 120585119905 where 120585

119905= 119862sum

infin

119895=119905119903119895|119910119905minus119895|

for some 0 lt 119903 lt 1 Therefore

100381710038171003817100381711987021003817100381710038171003817 le

119862

ℎradic119899 minus 119906

119899

sum119905=119906+1

119908119905

100381710038171003817100381711987611990510038171003817100381710038171003816100381610038161003816120576119905 minus 120576119905 (1206010)

1003816100381610038161003816

+119862

radic119899 minus 119906

119899

sum119905=119906+1

119908119905

1003817100381710038171003817119876119905 minus 119860 119905 (1206010)1003817100381710038171003817

le119862

ℎradic119899 minus 119906

119899

sum119905=119906+1

120585119905+

119862

radic119899 minus 119906

119899

sum119905=119906+1

120585119905

=119862

ℎradic119899 minus 119906

119899

sum119905=119906+1

119903119905

infin

sum119897=0

119903119897 1003816100381610038161003816119910minus119897

1003816100381610038161003816

+119862

radic119899 minus 119906

119899

sum119905=119906+1

119903119905

infin

sum119897=0

119903119897 1003816100381610038161003816119910minus119897

1003816100381610038161003816as997888rarr 0

(24)

Thus1198702is 119874(radiclog 119899) as For119870

3 we have

100381710038171003817100381711987031003817100381710038171003817

le119862

radic119899 minus 119906

119899

sum119905=119906+1

1003816100381610038161003816119908119905 minus 11990811990510038161003816100381610038161003817100381710038171003817119860 119905 (1206010)

1003817100381710038171003817

le119862

radic119899 minus 119906

119899

sum119905=119906+1

suminfin

119896=119905119896minus120572

1003816100381610038161003816119910119905minus1198961003816100381610038161003816 sum119905minus1

119895=111990311989510038161003816100381610038161003816119910119905minus119895

10038161003816100381610038161003816

1 + sum119905minus1

119896=1119896minus120572

1003816100381610038161003816119910119905minus1198961003816100381610038161003816

le119862

radic119899 minus 119906

119899

sum119905=119906+1

infin

sum119896=119905

119896minus120572 1003816100381610038161003816119910119905minus119896

1003816100381610038161003816

119905minus1

sum119895=1

119903119895119895120572

le119862

radic119899 minus 119906

119899

sum119905=119906+1

infin

sum119896=119905

119896minus120572 1003816100381610038161003816119910119905minus119896

1003816100381610038161003816

le119862

radic119899 minus 119906

119899

sum119905=119906+1

infin

sum119897=0

(119905 + 119897)minus120572 1003816100381610038161003816119910minus119897

1003816100381610038161003816

le119862

radic119899 minus 119906

119899

sum119905=119906+1

(119905minus120572 10038161003816100381610038161199100

1003816100381610038161003816 +

infin

sum119897=1

2minus120572119905minus1205722

119897minus1205722 1003816100381610038161003816119910minus119897

1003816100381610038161003816)

=119862

radic119899 minus 119906

119899

sum119905=119906+1

119905minus120572 10038161003816100381610038161199100

1003816100381610038161003816

+119862

radic119899 minus 119906

119899

sum119905=119906+1

119905minus1205722

infin

sum119897=1

119897minus1205722 1003816100381610038161003816119910minus119897

1003816100381610038161003816as997888rarr 0

(25)

because we have the facts that 119860119905(1206010) le 119862sum

119905minus1

119895=1119903119895|119910119905minus119895| (see

[2]) and (119905+ 119897)minus120572 le 2minus120572(119905119897)minus1205722 for 119905 gt 0 119897 gt 0 Thus119870

3is also

119874(radiclog 119899) as Therefore part (i) holds For the proof of part(ii) we may write

120597119876119899ℎ(1206010)

120597120601120591

= minus1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119908119905119860119905(1206010) 119860120591

1119905(1206010)

times [119870(120576119905(1206010)

ℎ) + 119870(minus

120576119905(1206010)

ℎ)]

+1

119899 minus 119906

119899

sum119905=119906+1

119908119905

120597119860119905(1206010)

120597120601120591119866ℎ(120576119905(1206010))

= 1198631+ 1198632

(26)

where 1198601119905(120601) = minus120597120576

119905(120601)120597120601 For119863

1 we may write

1198631= minus

1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119908119905119876119905119876120591

1119905(119870(

120576119905

ℎ) + 119870(minus

120576119905

ℎ))

minus1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

[119908119905119860119905(1206010) 119860120591

1119905(1206010)

times (119870(120576119905(1206010)

ℎ) + 119870(minus

120576119905(1206010)

ℎ))

minus119908119905119876119905119876120591

1119905(119870(

120576119905

ℎ) + 119870(minus

120576119905

ℎ))]

= 11986311+ 11986312

(27)

Journal of Applied Mathematics 5

Note that

11986311= minus

1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119879119905119888119899119905minus

1

(119899 minus 119906) ℎ

times

119899

sum119905=119906+1

119908119905119876119905119876120591

1119905119864(119870(

120576119905

ℎ) + 119870(minus

120576119905

ℎ))

= minus1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119879119905119888119899119905

minus [1

119899 minus 119906

119899

sum119905=119906+1

119908119905119876119905119876120591

1119905] (2119891 (0) + 119900 (ℎ))

(28)

where 119879119905= 1199081199051198761199051198761205911119905and 119888119899119905= 119870(120576

119905ℎ) + 119870(minus120576

119905ℎ) minus 119864[119870(120576

119905

ℎ) + 119870(minus120576119905ℎ)] The second term of (28) is minus2119891(0)Σ

1as by

the ergodicity We will prove that the first term is 119900(1) asWe suppose that 119876

119905is the first element 119880

119905minus1without loss of

generality Note that for each 119899 ge 119906 + 1 119879119905119888119899119905F119905 119906 + 1 le

119905 le 119899 is a sequence ofmartingale differences with |119879119905119888119899119905| le 119862

and

1198812

119899=

119899

sum119905=119906+1

119864 (119879119905119888119899119905)2

| F119905minus1

=

119899

sum119905=119906+1

1198792

119905119864(119888119899119905)2

lt 119899119862 (119891 (0) 1198620ℎ + 119874 (ℎ

2)) as

(29)

where 1198620gt 0 is a constant Set 119910 = 119899119862(119891(0)119862

0ℎ + 119874(ℎ2)) by

Theorem 12A in [16] for all 120576 gt 0 we have

119875

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum119905=119906+1

119879119905119888119899119905

1003816100381610038161003816100381610038161003816100381610038161003816

gt (119899ℎ) 120576

= 119875

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum119905=119906+1

119879119905119888119899119905

1003816100381610038161003816100381610038161003816100381610038161003816

gt (119899ℎ) 120576 1198812

119899lt 119910 for some 119899

le 2 exp minus(119899ℎ)21205762

2 (119910 + 119862119899ℎ120576)

= 2 exp minus119899ℎ1205762

2119862 (119891 (0) 1198620+ 119874 (ℎ)) + 2119862120576

(30)

The result follows from the Borel-Cantelli lemmaThus11986311=

minus2119891(0)Σ1+ 119900(1) as Similar to119870

2and119870

3 we have

1003817100381710038171003817119863121003817100381710038171003817

le1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119908119905(1003817100381710038171003817119876119905 minus 119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198761119905

1003817100381710038171003817

+1003817100381710038171003817119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198761119905 minus 1198601119905 (1206010)

1003817100381710038171003817)

100381610038161003816100381610038161003816100381610038161003816119870(

120576119905(1206010)

ℎ) + 119870(minus

120576119905(1206010)

ℎ)

100381610038161003816100381610038161003816100381610038161003816

+1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

1003816100381610038161003816119908119905 minus 11990811990510038161003816100381610038161003817100381710038171003817119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198601119905 (1206010)

1003817100381710038171003817

times

100381610038161003816100381610038161003816100381610038161003816119870(

120576119905(1206010)

ℎ) + 119870(minus

120576119905(1206010)

ℎ)

100381610038161003816100381610038161003816100381610038161003816

+1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119908119905

1003817100381710038171003817119876119905100381710038171003817100381710038171003817100381710038171198761119905

1003817100381710038171003817

times

100381610038161003816100381610038161003816100381610038161003816[119870(

120576119905

ℎ) + 119870(minus

120576119905

ℎ)]

minus[119870(120576119905(1206010)

ℎ) + 119870(minus

120576119905(1206010)

ℎ)]

100381610038161003816100381610038161003816100381610038161003816

le119862

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119908119905(1003817100381710038171003817119876119905 minus 119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198761119905

1003817100381710038171003817

+1003817100381710038171003817119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198761119905 minus 1198601119905 (1206010)

1003817100381710038171003817)

+119862

(119899 minus 119906) ℎ

119899

sum119905=119906+1

1003816100381610038161003816119908119905 minus 11990811990510038161003816100381610038161003817100381710038171003817119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198601119905 (1206010)

1003817100381710038171003817

+119862

(119899 minus 119906) ℎ2

119899

sum119905=119906+1

119908119905

1003817100381710038171003817119876119905100381710038171003817100381710038171003817100381710038171198761119905

10038171003817100381710038171003816100381610038161003816120576119905 minus 120576119905 (1206010)

1003816100381610038161003816

le119862

(119899 minus 119906) ℎ

119899

sum119905=119906+1

120585119905+

119862

(119899 minus 119906) ℎ

119899

sum119905=119906+1

infin

sum119896=119905

119896minus120572 1003816100381610038161003816119910119905minus119896

1003816100381610038161003816

+119862

(119899 minus 119906) ℎ2

119899

sum119905=119906+1

120585119905

as997888rarr 0

(31)

Therefore 1198631

= minus2119891(0)Σ1+ 119900(1) as For 119863

2 from the

definition of 119860119905(120601) it holds for 119905 gt max(119901 119902) that

120579 (119861)120597119860119905119894(120601)

120597120601119895

= 0 119894 119895 = 1 119901

120579 (119861)120597119860119905119894(120601)

120597120601119895+119901

= minus119860119905minus119895119894

(120601) 119894 = 1 119901 119895 = 1 119902

120579 (119861)120597119860119905119895+119901

(120601)

120597120601119894

= minus119860119905minus119895119894

(120601) 119894 = 1 119901 119895 = 1 119902

120579 (119861)120597119860119905119894+119901

(120601)

120597120601119895+119901

= minus119860119905minus119895119894+119901

(120601) minus 119860119905minus119894119895+119901

(120601)

119894 119895 = 1 119902

(32)

6 Journal of Applied Mathematics

where 119861 is the backshift operator For 119905 = 0 plusmn1 plusmn2 define

1205790(119861)119883119905(119894119895)

= 0 119894 119895 = 1 119901

1205790(119861)119883119905(119894119895+119901)

= minus119876119905minus119895119894

119894 = 1 119901 119895 = 1 119902

1205790(119861)119883119905(119895+119901119894)

= minus119876119905minus119895119894

119894 = 1 119901 119895 = 1 119902

1205790(119861)119883119905(119894+119901119895+119901)

= minus119876119905minus119895119894+119901

minus 119876119905minus119894119895+119901

119894 119895 = 1 119902

(33)

where 119876119905119894

is the 119894th component of 119876119905 Put 119883

119905= (119883

119905(119894119895))

similar to [13] we have that 120597119860119905(1206010)120597120601120591 le 119862sum

119905minus1

119895=1119903119895|119910119905minus119895|

119883119905 le 119862sum

infin

119895=1119903119895|119910119905minus119895| and 119883

119905minus 120597119860119905(1206010)120597120601120591 le 120585

119905 Then

we may write

1198632=

1

119899 minus 119906

119899

sum119905=119906+1

119908119905119883119905119866ℎ(120576119905)

+1

119899 minus 119906

119899

sum119905=119906+1

[119908119905

120597119860119905(1206010)

120597120601120591119866ℎ(120576119905(1206010)) minus 119908

119905119883119905119866ℎ(120576119905)]

= 11986321+ 11986322

(34)

Similar to11986311and119863

12 we have that119863

21and119863

22are 119900(1) as

This completes the proof

Lemma 4 Under the conditions of Proposition 1 as 119899 rarr infin

(119894) 119885119899(120601) = 119900 (119899

13) as

(119894119894) 119878 (120601) = Ω + 119900 (1) as(35)

hold uniformly in 119861

Proof of Lemma 4 For part (i) from [2] we havethat 120597119860

119905(120601)120597120601120591 le 119862sum

119905minus1

119895=1119903119895|119910119905minus119895| and 119860

119905(120601) le

119862sum119905minus1

119895=1119903119895|119910119905minus119895| uniformly hold in the ball 119861 for sufficiently

large 119899 Then for each 120601 isin 119861 we have1003817100381710038171003817119898119905ℎ (120601)

1003817100381710038171003817 le 119908119905

1003817100381710038171003817119860 119905 (120601)1003817100381710038171003817

le119862sum119905minus1

119895=111990311989510038161003816100381610038161003816119910119905minus119895

10038161003816100381610038161003816

1 + sum119905minus1

119896=1119896minus120572

1003816100381610038161003816119910119905minus1198961003816100381610038161003816

le 119862

119905minus1

sum119896=1

119903119896119896120572

le 119862

infin

sum119896=1

119903119896119896120572lt infin

(36)

Thus part (i) holds For part (ii) similar to the proof ofLemma 3 we have that 119878(120601

0) = Ω + 119900(1) as For each 120601 isin 119861

by Taylor expansion we have

119878 (120601) minus 119878 (1206010)

=1

119899 minus 119906

119899

sum119905=119906+1

119898119905ℎ(1206010)

120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)

120591

+1

119899 minus 119906

119899

sum119905=119906+1

120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)119898120591

119905ℎ(1206010)

+1

119899 minus 119906

119899

sum119905=119906+1

120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)

times 120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)

120591

= 1198791+ 1198792+ 1198793

(37)

where 120601lowast lies between 1206010and 120601 For 119879

1 we have

100381710038171003817100381711987911003817100381710038171003817 le

1

119899 minus 119906

119899

sum119905=119906+1

1199082

119905

1003817100381710038171003817119860 119905 (1206010)1003817100381710038171003817

100381710038171003817100381710038171003817100381710038171003817

120597119860119905(120601lowast)

120597120601120591

100381710038171003817100381710038171003817100381710038171003817

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

+1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

1199082

119905

1003817100381710038171003817119860 119905 (1206010)10038171003817100381710038171003817100381710038171003817119860 119905 (120601

lowast)1003817100381710038171003817

times10038171003817100381710038171198601119905 (120601

lowast)10038171003817100381710038171003817100381710038171003817120601 minus 1206010

1003817100381710038171003817

times

100381610038161003816100381610038161003816100381610038161003816119870(

120576119905(120601lowast)

ℎ) + 119870(minus

120576119905(120601lowast)

ℎ)

100381610038161003816100381610038161003816100381610038161003816

le119862119889119899

119899 minus 119906

119899

sum119905=119906+1

(

119905minus1

sum119895=1

119903119895119895120572)

2

+119862119889119899

(119899 minus 119906) ℎ

119899

sum119905=119906+1

(

119905minus1

sum119895=1

119903119895119895120572)

3

as997888rarr 0

(38)

Similarly we have 1198792

as997888997888rarr 0 and 119879

3

as997888997888rarr 0 This completes the

proof

Proof of Proposition 1 For 120601 isin 119861 by Taylor expansion

119876119899ℎ(120601) = 119876

119899ℎ(1206010) +

120597119876119899ℎ(1206010)

120597120601120591(120601 minus 120601

0)

+1

2

119901+119902minus119898

sum119895119896=1

1205972119876119899ℎ(1206010)

120597120601119895120597120601119896

(120601119895minus 1206010

119895) (120601119896minus 1206010

119896)

+1

6

119901+119902minus119898

sum119895119896119897=1

1205973119876119899ℎ(120601lowast)

120597120601119895120597120601119896120597120601119897

times (120601119895minus 1206010

119895) (120601119896minus 1206010

119896) (120601119897minus 1206010

119897)

(39)

where 120601lowastlies between 120601

0and 120601 Note that the final term on

the right side of (39) can be written as

1

6

119901+119902minus119898

sum119895119896119897=1

[

[

(120601119895minus 1206010119895) (120601119896minus 1206010119896) (120601119897minus 1206010119897)

ℎ3]

]

times (ℎ3

119899 minus 119906

119899

sum119905=119906+1

1205973119898119905ℎ(120601lowast)

120597120601119895120597120601119896120597120601119897

)

(40)

Journal of Applied Mathematics 7

which is 119900(120575119899) as where 120575

119899= 120601 minus 120601

0 because 1198892

119899ℎ3 =

11198992120573minus3120574 rarr 0 and1003817100381710038171003817100381710038171003817100381710038171003817

ℎ3

119899 minus 119906

119899

sum119905=119906+1

1205973119898119905ℎ(120601lowast)

120597120601119895120597120601119896120597120601119897

1003817100381710038171003817100381710038171003817100381710038171003817

le 119862 (41)

The third term on the right side of (39) can be written as

1

2

119901+119902minus119898

sum119895119896=1

[

[

(120601119895minus 1206010

119895) (120601119896minus 1206010

119896)

ℎ]

]

times ℎ

119899 minus 119906

119899

sum119905=119906+1

1205972119898119905ℎ(1206010)

120597120601119895120597120601119896

(42)

which is also 119900(120575119899) as because 119889

119899ℎ = 1119899120573minus120574 rarr 0 and

ℎ

119899 minus 119906

119899

sum119905=119906+1

1205972119898119905ℎ(1206010)

120597120601119895120597120601119896

= 119900 (1) as (43)

by a similar proof of Lemma 3 Therefore

119876119899ℎ(120601) = 119876

119899ℎ(1206010) +

120597119876119899ℎ(1206010)

120597120601120591(120601 minus 120601

0) + 119900 (120575

119899) as

(44)

uniformly about 120601 isin 119861 Denote 120601 = 1206010+ 120583119889119899 for 120601 isin 120601

120601 minus 1206010 = 119889

119899 where 120583 = 1 Now we give a lower bound

for 119897ℎ(120601) on the surface of the ball Similar to [6] by Lemmas

3 and 4 we have

119897ℎ(120601)

= (119899 minus 119906)119876119899ℎ(120601)120591

119878(120601)minus1

119876119899ℎ(120601) + 119900 (119899

43minus3120573) as

= (119899 minus 119906)[

[

119874(radiclog 119899119899

)

+ (minus2119891 (0)) Σ1120583119889119899+ 119900 (119889

119899) ]

]

120591

Ωminus1

times [

[

119874(radiclog 119899119899

) + (minus2119891 (0)) Σ1120583119889119899+ 119900 (119889

119899)]

]

+ 119900 (11989943minus3120573

) as

ge (119888 minus 120576) 1198991minus2120573 as

(45)

where 119888 minus 120576 gt 0 and 119888 is the smallest eigenvalue of41198912(0)Σ120591

1Ωminus1Σ1 Similarly

119897ℎ(1206010)

= (119899 minus 119906)119876119899ℎ(1206010)120591

119878(1206010)minus1

119876119899ℎ(1206010) + 119900 (1) as

= 119874 (log 119899) as

(46)

Since 119897ℎ(120601) is a continuous function about 120601 as 120601 belongs to

the ball 119861 119897ℎ(120601) attains its minimum value at some point 120601

in the interior of this ball and 120601 satisfies 120597119897ℎ(120601)120597120601 = 0 it

follows that (12) holds This completes the proof

Proof of Theorem 2 Similar to the proof ofTheorem 2 of Qinand Lawless [17] we have

(

120601 minus 1206010

) = 119878minus1

119899(minus1198761119899(1206010 0) + 119900

119901(119899minus12)

119900119901(119899minus12

)) (47)

where

119878119899= (

1205971198761119899(1206010 0)

120597120582120591

1205971198761119899(1206010 0)

120597120601120591

1205971198762119899(1206010 0)

1205971205821205910

)119901

997888rarr (11987811

11987812

11987821

0)

= (minusΩ minus2119891 (0) Σ

1

minus2119891 (0) Σ120591

10

)

(48)

By the standard arguments in the proof of empirical likeli-hood (see [6]) we have

119897119901(1205960)

= minus (119899 minus 119906) (119876119899ℎ(1206010)120591

0) 119878minus1

119899(119876119899ℎ(1206010)120591

0)120591

+ 119900119901(1)

= (Ωminus12radic119899 minus 119906119876

119899ℎ(1206010))120591

times (119868 minus 41198912(0)Ωminus12

Σ1Δminus1Σ120591

1Ωminus12

)

times (Ωminus12radic119899 minus 119906119876

119899ℎ(1206010)) + 119900

119901(1)

(49)

where Δ = 41198912(0)Σ1205911Ωminus1Σ1 Sinceradic119899 minus 119906119876

119899ℎ(1206010)119889

997888rarr 119873(0Ω)

and

tr 41198912 (0)Ωminus12Σ1Δminus1Σ120591

1Ωminus12

= tr Δminus141198912 (0) Σ1205911Ωminus1Σ1 = 119901 + 119902 minus 119898

(50)

it follows that 119897119901(1205960)119889

997888rarr 1205942119898

4 Simulation Studies

We generated data from a simple ARMA(1 1) model 119910119905=

1205931119910119905minus1

+ 120576119905+ 1205991120576119905minus1

with119873(0 1) 1199052 and Cauchy innovation

distribution We set 119906 = 20 120572 = 3 and the true value(1205931 1205991) = (04 07) or (minus05 07) where 120593

1is the parameter

of interest The sample size 119899 = 50 100 150 200 and2000 replications are conducted in all cases We smooth theestimating equations using kernel

119870 (119909) =1

radic2120587120590119890minus119909221205902

(51)

8 Journal of Applied Mathematics

Table 1 The coverage probability of confidence intervals when 120576119905sim 119873(0 1)

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08818 08820 08822 08193 07875100 08898 08896 08897 08655 08431150 08926 08927 08932 08692 08395200 08983 08983 08986 08666 08363

119886 = 09

1205931= minus05

50 08888 08885 08892 08156 07813100 08967 08968 08973 08738 08448150 08943 08944 08946 08823 08574200 08972 08979 08977 08931 08692

119886 = 095

1205931= 04

50 09347 09350 09350 08724 08425100 09424 09430 09431 09123 08862150 09467 09471 09470 09157 08936200 09494 09494 09497 09160 08937

119886 = 095

1205931= minus05

50 09404 09404 09404 08705 08407100 09472 09474 09474 09134 08931150 09481 09479 09476 09248 09052200 09495 09495 09490 09326 09152

where 120590 = 01 which is the so-called Gaussian kernel Thecoverage probabilities of smoothed empirical likelihood con-fidence regions 119868

ℎ119888with the bandwidth ℎ = 1119899120574 are denot-

ed by EL(120574) where 120574 = 027 030 032 respectivelyAs another benchmark of the simulation experiments

we consider the confidence regions based on the asymptoticnormal distribution ofWLADEproposed by [3] To constructthe confidence regions we need to estimate 119891(0) Σ and ΩWe can estimate 119891(0) by

119891 (0) =1

119908119887119899(119899 minus 119906)

119899

sum119905=119906+1

119908119905

120576119905(120579)

119887119899

(52)

where (119909) = exp(minus119909)(1+exp(minus119909))2 is a kernel function on119877 and 119887

119899= 1119899] is a bandwidth

119908= (119899 minus 119906)

minus1sum119899

119905=119906+1119908119905 Σ

andΩ can be estimated respectively by

Σ =1

119899 minus 119906

119899

sum119905=119906+1

119908119905119876119905119876120591

119905

Ω =1

119899 minus 119906

119899

sum119905=119906+1

1199082

119905119876119905119876120591

119905

(53)

where 119876119905is defined in the same manner as 119876

119905 1205790is replaced

by 120579 and 120576119905is replaced by 120576

119905(120579) see (14) Based on this we can

construct a NA confidence region (ie based on the normalapproximation of WLADE) The coverage probabilities ofconfidence regions 119868NA based on the bandwidth 119887

119899= 1119899] are

denoted byNA(]) with ] = 025 020 respectively Tables 12

and 3 show the probabilities of the confidence intervals of 1205931

at confidence levels 09 and 095 respectivelyThe simulation results can be summarized as followsThe

coverage probabilities of NA(]) are much smaller than thenominal levels and very sensitive to the choice of bandwidth119887119899and 120576119905 On the other hand the coverage probabilities of

EL(120574) are much better and less sensitive to the choice ofbandwidth ℎ and 120576

119905 As the sample size 119899 increases the

coverage probabilities for both increase to the nominal levelsas one might expect

5 Conclusions

This paper explores a profile empirical likelihood method toconstruct confidence regions for the partial parameters ofinterest in IVARMAmodels We started with the foundationof estimating equations of WLADE then from there wederived smoothed empirical likelihood Moreover we haveproved that the resulting statistics has asymptotic standardchi-squared distribution Hence there is no need to estimateany additional quantity such as the asymptotic variance Thesimulations indeed show that the proposed method has agood finite sample behavior which experimentally confirmsour method

Conflict of Interests

The authors declare that they have no conflict of interests

Journal of Applied Mathematics 9

Table 2 The coverage probability of confidence intervals when 120576119905sim 1199052

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08774 08776 08779 07627 07290100 08850 08849 08856 08323 07995150 08919 08917 08924 08467 08157200 08935 08932 08932 08510 08229

119886 = 09

1205931= minus05

50 08902 08902 08899 07295 07055100 08942 08944 08950 08151 07836150 08946 08941 08937 08473 08204200 08965 08963 08961 08641 08392

119886 = 095

1205931= 04

50 09331 09327 09327 08163 07854100 09412 09412 09413 08834 08542150 09447 09446 09447 08965 08718200 09456 09458 09459 09001 08762

119886 = 095

1205931= minus05

50 09421 09418 09418 07963 07658100 09455 09462 09460 08695 08430150 09444 09442 09440 08910 08695200 09464 09463 09464 09072 08863

Table 3 The coverage probability of confidence intervals when 120576119905sim 119862119886119906119888ℎ119910

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08360 08358 08361 06345 06018100 08708 08708 08702 07286 06942150 08811 08819 08823 07614 07270200 08870 08864 08868 07800 07467

119886 = 09

1205931= minus05

50 08673 08679 08678 05818 05502100 08869 08870 08874 06756 06440150 08972 08974 08974 07232 06867200 08956 08955 08953 07510 07156

119886 = 095

1205931= 04

50 08992 08994 08996 06888 06557100 09263 09261 09263 07870 07530150 09362 09366 09367 08176 07857200 09402 09398 09399 08350 08044

119886 = 095

1205931= minus05

50 09235 09239 09240 06378 06065100 09422 09420 09420 07342 07030150 09474 09479 09481 07820 07530200 09467 09471 09470 08057 07778

Acknowledgments

Lirsquos research is partially supported by the FundamentalResearch Funds for the Central Universities (no 2013XK03)Hersquos work is partially supported by the National NaturalScience Foundation of China (nos 11171230 and 11231010)

References

[1] TMikosch TGadrich C Kluppelberg andR J Adler ldquoParam-eter estimation for ARMA models with infinite variance inno-vationsrdquoTheAnnals of Statistics vol 23 no 1 pp 305ndash326 1995

[2] R A Davis ldquoGauss-Newton and M-estimation for ARMAprocesses with infinite variancerdquo Stochastic Processes and theirApplications vol 63 no 1 pp 75ndash95 1996

[3] J Pan HWang andQ Yao ldquoWeighted least absolute deviationsestimation forARMAmodels with infinite variancerdquoEconomet-ric Theory vol 23 no 5 pp 852ndash879 2007

[4] K Zhu and S Ling ldquoThe global LAD estimators for finiteinfinite variance ARMA(pq) modelsrdquo Econometric Theory vol28 no 5 pp 1065ndash1086 2012

[5] A B Owen ldquoEmpirical likelihood ratio confidence intervals fora single functionalrdquo Biometrika vol 75 no 2 pp 237ndash249 1988

[6] A BOwen ldquoEmpirical likelihood ratio confidence regionsrdquoTheAnnals of Statistics vol 18 no 1 pp 90ndash120 1990

10 Journal of Applied Mathematics

[7] A B Owen Empirical Likelihood Chapman and Hall LondonUK 2001

[8] A C Monti ldquoEmpirical likelihood confidence regions in timeseries modelsrdquo Biometrika vol 84 no 2 pp 395ndash405 1997

[9] C Chuang and N H Chan ldquoEmpirical likelihood for autore-gressive models with applications to unstable time seriesrdquoStatistica Sinica vol 12 no 2 pp 387ndash407 2002

[10] N H Chan L Peng and Y Qi ldquoQuantile inference for near-integrated autoregressive time series with infinite variancerdquoStatistica Sinica vol 16 no 1 pp 15ndash28 2006

[11] J Li W Liang S He and X Wu ldquoEmpirical likelihood forthe smoothed LAD estimator in infinite variance autoregressivemodelsrdquo Statistics amp Probability Letters vol 80 no 17-18 pp1420ndash1430 2010

[12] J Li W Liang and S He ldquoEmpirical likelihood for LADestimators in infinite variance ARMA modelsrdquo Statistics ampProbability Letters vol 81 no 2 pp 212ndash219 2011

[13] P J Brockwell and R A DavisTime seriesTheory andMethodsSpringer-Verlag New York NY USA 2nd edition 1991

[14] P C B Phillips ldquoA shortcut to LAD estimator asymptoticsrdquoEconometric Theory vol 7 no 4 pp 450ndash463 1991

[15] B W Silverman Density Estimation for Statistics and DataAnalysis Chapman ampHall London UK 1986

[16] V H de la Pena ldquoA general class of exponential inequalities formartingales and ratiosrdquoThe Annals of Probability vol 27 no 1pp 537ndash564 1999

[17] J Qin and J Lawless ldquoEmpirical likelihood and general estimat-ing equationsrdquo The Annals of Statistics vol 22 no 1 pp 300ndash325 1994

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

Page 3: Research Article Empirical Likelihood for Partial ...downloads.hindawi.com/journals/jam/2014/868970.pdf · Research Article Empirical Likelihood for Partial Parameters in ARMA Models

Journal of Applied Mathematics 3

Proposition 1 Let 119889119899= 1119899120573 with max13 31205742 lt 120573 lt

12 Assume (A1)ndash(A5) hold then as 119899 rarr infin with probability1 there exists a local minimizer 120601 of 119897

ℎ(120601 1205960) which lies in the

interior of the ball 119861 = 120601 120601 minus 1206010 le 119889

119899 Moreover 120601 and

= 120582(120601 1205960) satisfy

1198761119899(120601 ) = 0 119876

2119899(120601 ) = 0 (12)

where

1198761119899(120601 120582) =

1

119899 minus 119906

119899

sum119905=119906+1

119898119905ℎ(120601 1205960)

1 + 120582120591119898119905ℎ(120601 1205960)

1198762119899(120601 120582)

=1

119899 minus 119906

119899

sum119905=119906+1

1

1 + 120582120591119898119905ℎ(120601 1205960)(120597119898119905ℎ(120601 1205960)

120597120601120591)

120591

120582

(13)

The following theorem presents the asymptotic distribu-tion of the profile empirical likelihood

Theorem 2 Under conditions of Proposition 1 as 119899 rarr infinthe random variable 119897

119901(1205960) with 120601 given in Proposition 1

converges in distribution to 1205942119898

If 119888 is chosen such that 119875(1205942119898le 119888) = 119886 then Theorem 2

implies that the asymptotic coverage probability of empiricallikelihood confidence region 119868

ℎ119888= (120596 119897

119901(120596) le 119888) will be 119886

that is 119875(1205960isin 119868ℎ119888) = 119875(119897

119901(1205960) le 119888) = 119886 + 119900(1) as 119899 rarr infin

3 Proofs of the Main Results

In the following sdot denotes the Euclidian norm for a vectoror matrix and 119862 denotes a positive constant which may bedifferent at different places For 119905 = 0 plusmn1 plusmn2 define

119880119905minus

119901

sum119894=1

1205930

119894119880119905minus119894

= 120576119905 119881

119905+

119902

sum119895=1

1205990

119895119881119905minus119895

= 120576119905 (14)

Put 119876119905

= (119880119905minus1 119880

119905minus119901 119881119905minus1 119881

119905minus119902)120591 119908119905

= 1(1 +

suminfin

119896=1119896minus120572|119910119905minus119896|)4 and the corresponding partial vector for 120601

0

is denoted by 1198761119905 Let

Σ = 119864 (119908119905119876119905119876120591

119905)

Σ1= 119864 (119908

119905119876119905119876120591

1119905)

Ω = 119864 (1199082

119905119876119905119876120591

119905)

(15)

Assumptions A1 and A2 imply that for 120575 = min(120575 1)

119864(

infin

sum119896=1

119896minus1205722 1003816100381610038161003816119910119905minus119896

1003816100381610038161003816)

120575

le

infin

sum119896=1

119896minus1205721205752

1198641003816100381610038161003816119910119905minus119896

1003816100381610038161003816120575

lt infin (16)

Hence suminfin119896=1

119896minus1205722|119910119905minus119896| lt infin with probability 1 which

ensures that 119908119905

is well defined Note that 119876119905 le

119862suminfin

119895=1119903119895|119910119905minus119895| for some 0 lt 119903 lt 1 and

11990812

119905

10038171003817100381710038171198761199051003817100381710038171003817 le

119862suminfin

119895=111990311989510038161003816100381610038161003816119910119905minus119895

10038161003816100381610038161003816

1 + suminfin

119896=1119896minus120572

1003816100381610038161003816119910119905minus1198961003816100381610038161003816le 119862

infin

sum119896=1

119903119896119896120572lt infin (17)

Then Σ Σ1 and Ω are well-defined (finite) matrices For

simplicity we denote (120601 1205960) and (120601

0 1205960) by 120601 and 120601

0

respectively in this section The following notations will beused in the proofs Let

119885119899(120601) = max

119906+1le119905le119899

1003817100381710038171003817119898119905ℎ (120601)1003817100381710038171003817

119876119899ℎ(120601) =

1

119899 minus 119906

119899

sum119905=119906+1

119898119905ℎ(120601)

119878 (120601) =1

119899 minus 119906

119899

sum119905=119906+1

119898119905ℎ(120601)119898

119905ℎ(120601)120591

(18)

To prove Proposition 1 we first prove the following lemmas

Lemma 3 Under the conditions of Proposition 1 as 119899 rarr infin

(119894) 119876119899ℎ(1206010) = 119874(radic

log 119899119899

) as

(119894119894)120597119876119899ℎ(1206010)

120597120601120591= minus2119891 (0) Σ

1+ 119900 (1) as

(19)

Proof of Lemma 3 For part (i) we may write

radic119899 minus 119906119876119899ℎ(1206010)

=1

radic119899 minus 119906

119899

sum119905=119906+1

119908119905119876119905119866ℎ(120576119905)

+1

radic119899 minus 119906

119899

sum119905=119906+1

[119908119905119876119905(119866ℎ(120576119905(1206010)) minus 119866

ℎ(120576119905))

+119908119905(119860119905(1206010) minus 119876119905) 119866ℎ(120576119905(1206010)) ]

+1

radic119899 minus 119906

119899

sum119905=119906+1

(119908119905minus 119908119905) 119860119905(1206010) 119866ℎ(120576119905(1206010))

= 1198701+ 1198702+ 1198703

(20)

For1198701 we have

1198701

radic119899 minus 119906=

1

119899 minus 119906

119899

sum119905=119906+1

119885119905119887119899119905

+ (1

119899 minus 119906

119899

sum119905=119906+1

119885119905)119874(ℎ

2)

(21)

where 119885119905= 119908119905119876119905 119887119899119905= 119866ℎ(120576119905) minus 119864(119866

ℎ(120576119905)) The second term

of (21) is 119874(radiclog 119899119899) as by the ergodicity Now turning tothe first term we suppose that 119876

119905is the first element 119880

119905minus1

without loss of generality Note that for each 119899 ge 119906 + 1119885119905119887119899119905F119905 119906 + 1 le 119905 le 119899 is a sequence of martingale

4 Journal of Applied Mathematics

differences with |119885119905119887119899119905| le 119862 where F

119905= 120590(120576

119904 119904 le 119905) For

some 1198620gt 0 by the ergodicity we have

1198812

119899=

119899

sum119905=119906+1

119864 (119885119905119887119899119905)2

| F119905minus1

=

119899

sum119905=119906+1

1198852

119905119864(119887119899119905)2

le 119862

119899

sum119905=119906+1

1198852

119905lt 119862 (119864 (119885

2

119905) + 1198620) 119899 as

(22)

Set119910 = 119862(119864(1198852119905)+1198620)119899 byTheorem 12A in [16] for all119860 gt 0

we have

119875

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum119905=119906+1

119885119905119887119899119905

1003816100381610038161003816100381610038161003816100381610038161003816

gt 119860radic119899 log 119899

= 119875

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum119905=119906+1

119885119905119887119899119905

1003816100381610038161003816100381610038161003816100381610038161003816

gt 119860radic119899 log 119899 1198812119899lt 119910 for some 119899

le 2 expminus1198602119899 log 119899

2 (119910 + 119862119860radic119899 log 119899)

= 2 expminus1198602 log 119899

2119862 (119864 (1198852119905) + 1198620) + 2119862119860radiclog 119899119899

(23)

Choosing 119860 such that 1198602 gt 2119862(119864(1198852119905) + 1198620) by the Borel-

Cantelli lemma the first term of (21) is119874(radiclog 119899119899) asThus1198701is 119874(radiclog 119899) as For 119870

2 by Davis [2] it holds that |120576

119905minus

120576119905(1206010)| le 120585119905 and 119860

119905(1206010)minus119876119905 le 120585119905 where 120585

119905= 119862sum

infin

119895=119905119903119895|119910119905minus119895|

for some 0 lt 119903 lt 1 Therefore

100381710038171003817100381711987021003817100381710038171003817 le

119862

ℎradic119899 minus 119906

119899

sum119905=119906+1

119908119905

100381710038171003817100381711987611990510038171003817100381710038171003816100381610038161003816120576119905 minus 120576119905 (1206010)

1003816100381610038161003816

+119862

radic119899 minus 119906

119899

sum119905=119906+1

119908119905

1003817100381710038171003817119876119905 minus 119860 119905 (1206010)1003817100381710038171003817

le119862

ℎradic119899 minus 119906

119899

sum119905=119906+1

120585119905+

119862

radic119899 minus 119906

119899

sum119905=119906+1

120585119905

=119862

ℎradic119899 minus 119906

119899

sum119905=119906+1

119903119905

infin

sum119897=0

119903119897 1003816100381610038161003816119910minus119897

1003816100381610038161003816

+119862

radic119899 minus 119906

119899

sum119905=119906+1

119903119905

infin

sum119897=0

119903119897 1003816100381610038161003816119910minus119897

1003816100381610038161003816as997888rarr 0

(24)

Thus1198702is 119874(radiclog 119899) as For119870

3 we have

100381710038171003817100381711987031003817100381710038171003817

le119862

radic119899 minus 119906

119899

sum119905=119906+1

1003816100381610038161003816119908119905 minus 11990811990510038161003816100381610038161003817100381710038171003817119860 119905 (1206010)

1003817100381710038171003817

le119862

radic119899 minus 119906

119899

sum119905=119906+1

suminfin

119896=119905119896minus120572

1003816100381610038161003816119910119905minus1198961003816100381610038161003816 sum119905minus1

119895=111990311989510038161003816100381610038161003816119910119905minus119895

10038161003816100381610038161003816

1 + sum119905minus1

119896=1119896minus120572

1003816100381610038161003816119910119905minus1198961003816100381610038161003816

le119862

radic119899 minus 119906

119899

sum119905=119906+1

infin

sum119896=119905

119896minus120572 1003816100381610038161003816119910119905minus119896

1003816100381610038161003816

119905minus1

sum119895=1

119903119895119895120572

le119862

radic119899 minus 119906

119899

sum119905=119906+1

infin

sum119896=119905

119896minus120572 1003816100381610038161003816119910119905minus119896

1003816100381610038161003816

le119862

radic119899 minus 119906

119899

sum119905=119906+1

infin

sum119897=0

(119905 + 119897)minus120572 1003816100381610038161003816119910minus119897

1003816100381610038161003816

le119862

radic119899 minus 119906

119899

sum119905=119906+1

(119905minus120572 10038161003816100381610038161199100

1003816100381610038161003816 +

infin

sum119897=1

2minus120572119905minus1205722

119897minus1205722 1003816100381610038161003816119910minus119897

1003816100381610038161003816)

=119862

radic119899 minus 119906

119899

sum119905=119906+1

119905minus120572 10038161003816100381610038161199100

1003816100381610038161003816

+119862

radic119899 minus 119906

119899

sum119905=119906+1

119905minus1205722

infin

sum119897=1

119897minus1205722 1003816100381610038161003816119910minus119897

1003816100381610038161003816as997888rarr 0

(25)

because we have the facts that 119860119905(1206010) le 119862sum

119905minus1

119895=1119903119895|119910119905minus119895| (see

[2]) and (119905+ 119897)minus120572 le 2minus120572(119905119897)minus1205722 for 119905 gt 0 119897 gt 0 Thus119870

3is also

119874(radiclog 119899) as Therefore part (i) holds For the proof of part(ii) we may write

120597119876119899ℎ(1206010)

120597120601120591

= minus1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119908119905119860119905(1206010) 119860120591

1119905(1206010)

times [119870(120576119905(1206010)

ℎ) + 119870(minus

120576119905(1206010)

ℎ)]

+1

119899 minus 119906

119899

sum119905=119906+1

119908119905

120597119860119905(1206010)

120597120601120591119866ℎ(120576119905(1206010))

= 1198631+ 1198632

(26)

where 1198601119905(120601) = minus120597120576

119905(120601)120597120601 For119863

1 we may write

1198631= minus

1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119908119905119876119905119876120591

1119905(119870(

120576119905

ℎ) + 119870(minus

120576119905

ℎ))

minus1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

[119908119905119860119905(1206010) 119860120591

1119905(1206010)

times (119870(120576119905(1206010)

ℎ) + 119870(minus

120576119905(1206010)

ℎ))

minus119908119905119876119905119876120591

1119905(119870(

120576119905

ℎ) + 119870(minus

120576119905

ℎ))]

= 11986311+ 11986312

(27)

Journal of Applied Mathematics 5

Note that

11986311= minus

1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119879119905119888119899119905minus

1

(119899 minus 119906) ℎ

times

119899

sum119905=119906+1

119908119905119876119905119876120591

1119905119864(119870(

120576119905

ℎ) + 119870(minus

120576119905

ℎ))

= minus1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119879119905119888119899119905

minus [1

119899 minus 119906

119899

sum119905=119906+1

119908119905119876119905119876120591

1119905] (2119891 (0) + 119900 (ℎ))

(28)

where 119879119905= 1199081199051198761199051198761205911119905and 119888119899119905= 119870(120576

119905ℎ) + 119870(minus120576

119905ℎ) minus 119864[119870(120576

119905

ℎ) + 119870(minus120576119905ℎ)] The second term of (28) is minus2119891(0)Σ

1as by

the ergodicity We will prove that the first term is 119900(1) asWe suppose that 119876

119905is the first element 119880

119905minus1without loss of

generality Note that for each 119899 ge 119906 + 1 119879119905119888119899119905F119905 119906 + 1 le

119905 le 119899 is a sequence ofmartingale differences with |119879119905119888119899119905| le 119862

and

1198812

119899=

119899

sum119905=119906+1

119864 (119879119905119888119899119905)2

| F119905minus1

=

119899

sum119905=119906+1

1198792

119905119864(119888119899119905)2

lt 119899119862 (119891 (0) 1198620ℎ + 119874 (ℎ

2)) as

(29)

where 1198620gt 0 is a constant Set 119910 = 119899119862(119891(0)119862

0ℎ + 119874(ℎ2)) by

Theorem 12A in [16] for all 120576 gt 0 we have

119875

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum119905=119906+1

119879119905119888119899119905

1003816100381610038161003816100381610038161003816100381610038161003816

gt (119899ℎ) 120576

= 119875

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum119905=119906+1

119879119905119888119899119905

1003816100381610038161003816100381610038161003816100381610038161003816

gt (119899ℎ) 120576 1198812

119899lt 119910 for some 119899

le 2 exp minus(119899ℎ)21205762

2 (119910 + 119862119899ℎ120576)

= 2 exp minus119899ℎ1205762

2119862 (119891 (0) 1198620+ 119874 (ℎ)) + 2119862120576

(30)

The result follows from the Borel-Cantelli lemmaThus11986311=

minus2119891(0)Σ1+ 119900(1) as Similar to119870

2and119870

3 we have

1003817100381710038171003817119863121003817100381710038171003817

le1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119908119905(1003817100381710038171003817119876119905 minus 119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198761119905

1003817100381710038171003817

+1003817100381710038171003817119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198761119905 minus 1198601119905 (1206010)

1003817100381710038171003817)

100381610038161003816100381610038161003816100381610038161003816119870(

120576119905(1206010)

ℎ) + 119870(minus

120576119905(1206010)

ℎ)

100381610038161003816100381610038161003816100381610038161003816

+1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

1003816100381610038161003816119908119905 minus 11990811990510038161003816100381610038161003817100381710038171003817119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198601119905 (1206010)

1003817100381710038171003817

times

100381610038161003816100381610038161003816100381610038161003816119870(

120576119905(1206010)

ℎ) + 119870(minus

120576119905(1206010)

ℎ)

100381610038161003816100381610038161003816100381610038161003816

+1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119908119905

1003817100381710038171003817119876119905100381710038171003817100381710038171003817100381710038171198761119905

1003817100381710038171003817

times

100381610038161003816100381610038161003816100381610038161003816[119870(

120576119905

ℎ) + 119870(minus

120576119905

ℎ)]

minus[119870(120576119905(1206010)

ℎ) + 119870(minus

120576119905(1206010)

ℎ)]

100381610038161003816100381610038161003816100381610038161003816

le119862

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119908119905(1003817100381710038171003817119876119905 minus 119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198761119905

1003817100381710038171003817

+1003817100381710038171003817119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198761119905 minus 1198601119905 (1206010)

1003817100381710038171003817)

+119862

(119899 minus 119906) ℎ

119899

sum119905=119906+1

1003816100381610038161003816119908119905 minus 11990811990510038161003816100381610038161003817100381710038171003817119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198601119905 (1206010)

1003817100381710038171003817

+119862

(119899 minus 119906) ℎ2

119899

sum119905=119906+1

119908119905

1003817100381710038171003817119876119905100381710038171003817100381710038171003817100381710038171198761119905

10038171003817100381710038171003816100381610038161003816120576119905 minus 120576119905 (1206010)

1003816100381610038161003816

le119862

(119899 minus 119906) ℎ

119899

sum119905=119906+1

120585119905+

119862

(119899 minus 119906) ℎ

119899

sum119905=119906+1

infin

sum119896=119905

119896minus120572 1003816100381610038161003816119910119905minus119896

1003816100381610038161003816

+119862

(119899 minus 119906) ℎ2

119899

sum119905=119906+1

120585119905

as997888rarr 0

(31)

Therefore 1198631

= minus2119891(0)Σ1+ 119900(1) as For 119863

2 from the

definition of 119860119905(120601) it holds for 119905 gt max(119901 119902) that

120579 (119861)120597119860119905119894(120601)

120597120601119895

= 0 119894 119895 = 1 119901

120579 (119861)120597119860119905119894(120601)

120597120601119895+119901

= minus119860119905minus119895119894

(120601) 119894 = 1 119901 119895 = 1 119902

120579 (119861)120597119860119905119895+119901

(120601)

120597120601119894

= minus119860119905minus119895119894

(120601) 119894 = 1 119901 119895 = 1 119902

120579 (119861)120597119860119905119894+119901

(120601)

120597120601119895+119901

= minus119860119905minus119895119894+119901

(120601) minus 119860119905minus119894119895+119901

(120601)

119894 119895 = 1 119902

(32)

6 Journal of Applied Mathematics

where 119861 is the backshift operator For 119905 = 0 plusmn1 plusmn2 define

1205790(119861)119883119905(119894119895)

= 0 119894 119895 = 1 119901

1205790(119861)119883119905(119894119895+119901)

= minus119876119905minus119895119894

119894 = 1 119901 119895 = 1 119902

1205790(119861)119883119905(119895+119901119894)

= minus119876119905minus119895119894

119894 = 1 119901 119895 = 1 119902

1205790(119861)119883119905(119894+119901119895+119901)

= minus119876119905minus119895119894+119901

minus 119876119905minus119894119895+119901

119894 119895 = 1 119902

(33)

where 119876119905119894

is the 119894th component of 119876119905 Put 119883

119905= (119883

119905(119894119895))

similar to [13] we have that 120597119860119905(1206010)120597120601120591 le 119862sum

119905minus1

119895=1119903119895|119910119905minus119895|

119883119905 le 119862sum

infin

119895=1119903119895|119910119905minus119895| and 119883

119905minus 120597119860119905(1206010)120597120601120591 le 120585

119905 Then

we may write

1198632=

1

119899 minus 119906

119899

sum119905=119906+1

119908119905119883119905119866ℎ(120576119905)

+1

119899 minus 119906

119899

sum119905=119906+1

[119908119905

120597119860119905(1206010)

120597120601120591119866ℎ(120576119905(1206010)) minus 119908

119905119883119905119866ℎ(120576119905)]

= 11986321+ 11986322

(34)

Similar to11986311and119863

12 we have that119863

21and119863

22are 119900(1) as

This completes the proof

Lemma 4 Under the conditions of Proposition 1 as 119899 rarr infin

(119894) 119885119899(120601) = 119900 (119899

13) as

(119894119894) 119878 (120601) = Ω + 119900 (1) as(35)

hold uniformly in 119861

Proof of Lemma 4 For part (i) from [2] we havethat 120597119860

119905(120601)120597120601120591 le 119862sum

119905minus1

119895=1119903119895|119910119905minus119895| and 119860

119905(120601) le

119862sum119905minus1

119895=1119903119895|119910119905minus119895| uniformly hold in the ball 119861 for sufficiently

large 119899 Then for each 120601 isin 119861 we have1003817100381710038171003817119898119905ℎ (120601)

1003817100381710038171003817 le 119908119905

1003817100381710038171003817119860 119905 (120601)1003817100381710038171003817

le119862sum119905minus1

119895=111990311989510038161003816100381610038161003816119910119905minus119895

10038161003816100381610038161003816

1 + sum119905minus1

119896=1119896minus120572

1003816100381610038161003816119910119905minus1198961003816100381610038161003816

le 119862

119905minus1

sum119896=1

119903119896119896120572

le 119862

infin

sum119896=1

119903119896119896120572lt infin

(36)

Thus part (i) holds For part (ii) similar to the proof ofLemma 3 we have that 119878(120601

0) = Ω + 119900(1) as For each 120601 isin 119861

by Taylor expansion we have

119878 (120601) minus 119878 (1206010)

=1

119899 minus 119906

119899

sum119905=119906+1

119898119905ℎ(1206010)

120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)

120591

+1

119899 minus 119906

119899

sum119905=119906+1

120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)119898120591

119905ℎ(1206010)

+1

119899 minus 119906

119899

sum119905=119906+1

120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)

times 120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)

120591

= 1198791+ 1198792+ 1198793

(37)

where 120601lowast lies between 1206010and 120601 For 119879

1 we have

100381710038171003817100381711987911003817100381710038171003817 le

1

119899 minus 119906

119899

sum119905=119906+1

1199082

119905

1003817100381710038171003817119860 119905 (1206010)1003817100381710038171003817

100381710038171003817100381710038171003817100381710038171003817

120597119860119905(120601lowast)

120597120601120591

100381710038171003817100381710038171003817100381710038171003817

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

+1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

1199082

119905

1003817100381710038171003817119860 119905 (1206010)10038171003817100381710038171003817100381710038171003817119860 119905 (120601

lowast)1003817100381710038171003817

times10038171003817100381710038171198601119905 (120601

lowast)10038171003817100381710038171003817100381710038171003817120601 minus 1206010

1003817100381710038171003817

times

100381610038161003816100381610038161003816100381610038161003816119870(

120576119905(120601lowast)

ℎ) + 119870(minus

120576119905(120601lowast)

ℎ)

100381610038161003816100381610038161003816100381610038161003816

le119862119889119899

119899 minus 119906

119899

sum119905=119906+1

(

119905minus1

sum119895=1

119903119895119895120572)

2

+119862119889119899

(119899 minus 119906) ℎ

119899

sum119905=119906+1

(

119905minus1

sum119895=1

119903119895119895120572)

3

as997888rarr 0

(38)

Similarly we have 1198792

as997888997888rarr 0 and 119879

3

as997888997888rarr 0 This completes the

proof

Proof of Proposition 1 For 120601 isin 119861 by Taylor expansion

119876119899ℎ(120601) = 119876

119899ℎ(1206010) +

120597119876119899ℎ(1206010)

120597120601120591(120601 minus 120601

0)

+1

2

119901+119902minus119898

sum119895119896=1

1205972119876119899ℎ(1206010)

120597120601119895120597120601119896

(120601119895minus 1206010

119895) (120601119896minus 1206010

119896)

+1

6

119901+119902minus119898

sum119895119896119897=1

1205973119876119899ℎ(120601lowast)

120597120601119895120597120601119896120597120601119897

times (120601119895minus 1206010

119895) (120601119896minus 1206010

119896) (120601119897minus 1206010

119897)

(39)

where 120601lowastlies between 120601

0and 120601 Note that the final term on

the right side of (39) can be written as

1

6

119901+119902minus119898

sum119895119896119897=1

[

[

(120601119895minus 1206010119895) (120601119896minus 1206010119896) (120601119897minus 1206010119897)

ℎ3]

]

times (ℎ3

119899 minus 119906

119899

sum119905=119906+1

1205973119898119905ℎ(120601lowast)

120597120601119895120597120601119896120597120601119897

)

(40)

Journal of Applied Mathematics 7

which is 119900(120575119899) as where 120575

119899= 120601 minus 120601

0 because 1198892

119899ℎ3 =

11198992120573minus3120574 rarr 0 and1003817100381710038171003817100381710038171003817100381710038171003817

ℎ3

119899 minus 119906

119899

sum119905=119906+1

1205973119898119905ℎ(120601lowast)

120597120601119895120597120601119896120597120601119897

1003817100381710038171003817100381710038171003817100381710038171003817

le 119862 (41)

The third term on the right side of (39) can be written as

1

2

119901+119902minus119898

sum119895119896=1

[

[

(120601119895minus 1206010

119895) (120601119896minus 1206010

119896)

ℎ]

]

times ℎ

119899 minus 119906

119899

sum119905=119906+1

1205972119898119905ℎ(1206010)

120597120601119895120597120601119896

(42)

which is also 119900(120575119899) as because 119889

119899ℎ = 1119899120573minus120574 rarr 0 and

ℎ

119899 minus 119906

119899

sum119905=119906+1

1205972119898119905ℎ(1206010)

120597120601119895120597120601119896

= 119900 (1) as (43)

by a similar proof of Lemma 3 Therefore

119876119899ℎ(120601) = 119876

119899ℎ(1206010) +

120597119876119899ℎ(1206010)

120597120601120591(120601 minus 120601

0) + 119900 (120575

119899) as

(44)

uniformly about 120601 isin 119861 Denote 120601 = 1206010+ 120583119889119899 for 120601 isin 120601

120601 minus 1206010 = 119889

119899 where 120583 = 1 Now we give a lower bound

for 119897ℎ(120601) on the surface of the ball Similar to [6] by Lemmas

3 and 4 we have

119897ℎ(120601)

= (119899 minus 119906)119876119899ℎ(120601)120591

119878(120601)minus1

119876119899ℎ(120601) + 119900 (119899

43minus3120573) as

= (119899 minus 119906)[

[

119874(radiclog 119899119899

)

+ (minus2119891 (0)) Σ1120583119889119899+ 119900 (119889

119899) ]

]

120591

Ωminus1

times [

[

119874(radiclog 119899119899

) + (minus2119891 (0)) Σ1120583119889119899+ 119900 (119889

119899)]

]

+ 119900 (11989943minus3120573

) as

ge (119888 minus 120576) 1198991minus2120573 as

(45)

where 119888 minus 120576 gt 0 and 119888 is the smallest eigenvalue of41198912(0)Σ120591

1Ωminus1Σ1 Similarly

119897ℎ(1206010)

= (119899 minus 119906)119876119899ℎ(1206010)120591

119878(1206010)minus1

119876119899ℎ(1206010) + 119900 (1) as

= 119874 (log 119899) as

(46)

Since 119897ℎ(120601) is a continuous function about 120601 as 120601 belongs to

the ball 119861 119897ℎ(120601) attains its minimum value at some point 120601

in the interior of this ball and 120601 satisfies 120597119897ℎ(120601)120597120601 = 0 it

follows that (12) holds This completes the proof

Proof of Theorem 2 Similar to the proof ofTheorem 2 of Qinand Lawless [17] we have

(

120601 minus 1206010

) = 119878minus1

119899(minus1198761119899(1206010 0) + 119900

119901(119899minus12)

119900119901(119899minus12

)) (47)

where

119878119899= (

1205971198761119899(1206010 0)

120597120582120591

1205971198761119899(1206010 0)

120597120601120591

1205971198762119899(1206010 0)

1205971205821205910

)119901

997888rarr (11987811

11987812

11987821

0)

= (minusΩ minus2119891 (0) Σ

1

minus2119891 (0) Σ120591

10

)

(48)

By the standard arguments in the proof of empirical likeli-hood (see [6]) we have

119897119901(1205960)

= minus (119899 minus 119906) (119876119899ℎ(1206010)120591

0) 119878minus1

119899(119876119899ℎ(1206010)120591

0)120591

+ 119900119901(1)

= (Ωminus12radic119899 minus 119906119876

119899ℎ(1206010))120591

times (119868 minus 41198912(0)Ωminus12

Σ1Δminus1Σ120591

1Ωminus12

)

times (Ωminus12radic119899 minus 119906119876

119899ℎ(1206010)) + 119900

119901(1)

(49)

where Δ = 41198912(0)Σ1205911Ωminus1Σ1 Sinceradic119899 minus 119906119876

119899ℎ(1206010)119889

997888rarr 119873(0Ω)

and

tr 41198912 (0)Ωminus12Σ1Δminus1Σ120591

1Ωminus12

= tr Δminus141198912 (0) Σ1205911Ωminus1Σ1 = 119901 + 119902 minus 119898

(50)

it follows that 119897119901(1205960)119889

997888rarr 1205942119898

4 Simulation Studies

We generated data from a simple ARMA(1 1) model 119910119905=

1205931119910119905minus1

+ 120576119905+ 1205991120576119905minus1

with119873(0 1) 1199052 and Cauchy innovation

distribution We set 119906 = 20 120572 = 3 and the true value(1205931 1205991) = (04 07) or (minus05 07) where 120593

1is the parameter

of interest The sample size 119899 = 50 100 150 200 and2000 replications are conducted in all cases We smooth theestimating equations using kernel

119870 (119909) =1

radic2120587120590119890minus119909221205902

(51)

8 Journal of Applied Mathematics

Table 1 The coverage probability of confidence intervals when 120576119905sim 119873(0 1)

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08818 08820 08822 08193 07875100 08898 08896 08897 08655 08431150 08926 08927 08932 08692 08395200 08983 08983 08986 08666 08363

119886 = 09

1205931= minus05

50 08888 08885 08892 08156 07813100 08967 08968 08973 08738 08448150 08943 08944 08946 08823 08574200 08972 08979 08977 08931 08692

119886 = 095

1205931= 04

50 09347 09350 09350 08724 08425100 09424 09430 09431 09123 08862150 09467 09471 09470 09157 08936200 09494 09494 09497 09160 08937

119886 = 095

1205931= minus05

50 09404 09404 09404 08705 08407100 09472 09474 09474 09134 08931150 09481 09479 09476 09248 09052200 09495 09495 09490 09326 09152

where 120590 = 01 which is the so-called Gaussian kernel Thecoverage probabilities of smoothed empirical likelihood con-fidence regions 119868

ℎ119888with the bandwidth ℎ = 1119899120574 are denot-

ed by EL(120574) where 120574 = 027 030 032 respectivelyAs another benchmark of the simulation experiments

we consider the confidence regions based on the asymptoticnormal distribution ofWLADEproposed by [3] To constructthe confidence regions we need to estimate 119891(0) Σ and ΩWe can estimate 119891(0) by

119891 (0) =1

119908119887119899(119899 minus 119906)

119899

sum119905=119906+1

119908119905

120576119905(120579)

119887119899

(52)

where (119909) = exp(minus119909)(1+exp(minus119909))2 is a kernel function on119877 and 119887

119899= 1119899] is a bandwidth

119908= (119899 minus 119906)

minus1sum119899

119905=119906+1119908119905 Σ

andΩ can be estimated respectively by

Σ =1

119899 minus 119906

119899

sum119905=119906+1

119908119905119876119905119876120591

119905

Ω =1

119899 minus 119906

119899

sum119905=119906+1

1199082

119905119876119905119876120591

119905

(53)

where 119876119905is defined in the same manner as 119876

119905 1205790is replaced

by 120579 and 120576119905is replaced by 120576

119905(120579) see (14) Based on this we can

construct a NA confidence region (ie based on the normalapproximation of WLADE) The coverage probabilities ofconfidence regions 119868NA based on the bandwidth 119887

119899= 1119899] are

denoted byNA(]) with ] = 025 020 respectively Tables 12

and 3 show the probabilities of the confidence intervals of 1205931

at confidence levels 09 and 095 respectivelyThe simulation results can be summarized as followsThe

coverage probabilities of NA(]) are much smaller than thenominal levels and very sensitive to the choice of bandwidth119887119899and 120576119905 On the other hand the coverage probabilities of

EL(120574) are much better and less sensitive to the choice ofbandwidth ℎ and 120576

119905 As the sample size 119899 increases the

coverage probabilities for both increase to the nominal levelsas one might expect

5 Conclusions

This paper explores a profile empirical likelihood method toconstruct confidence regions for the partial parameters ofinterest in IVARMAmodels We started with the foundationof estimating equations of WLADE then from there wederived smoothed empirical likelihood Moreover we haveproved that the resulting statistics has asymptotic standardchi-squared distribution Hence there is no need to estimateany additional quantity such as the asymptotic variance Thesimulations indeed show that the proposed method has agood finite sample behavior which experimentally confirmsour method

Conflict of Interests

The authors declare that they have no conflict of interests

Journal of Applied Mathematics 9

Table 2 The coverage probability of confidence intervals when 120576119905sim 1199052

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08774 08776 08779 07627 07290100 08850 08849 08856 08323 07995150 08919 08917 08924 08467 08157200 08935 08932 08932 08510 08229

119886 = 09

1205931= minus05

50 08902 08902 08899 07295 07055100 08942 08944 08950 08151 07836150 08946 08941 08937 08473 08204200 08965 08963 08961 08641 08392

119886 = 095

1205931= 04

50 09331 09327 09327 08163 07854100 09412 09412 09413 08834 08542150 09447 09446 09447 08965 08718200 09456 09458 09459 09001 08762

119886 = 095

1205931= minus05

50 09421 09418 09418 07963 07658100 09455 09462 09460 08695 08430150 09444 09442 09440 08910 08695200 09464 09463 09464 09072 08863

Table 3 The coverage probability of confidence intervals when 120576119905sim 119862119886119906119888ℎ119910

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08360 08358 08361 06345 06018100 08708 08708 08702 07286 06942150 08811 08819 08823 07614 07270200 08870 08864 08868 07800 07467

119886 = 09

1205931= minus05

50 08673 08679 08678 05818 05502100 08869 08870 08874 06756 06440150 08972 08974 08974 07232 06867200 08956 08955 08953 07510 07156

119886 = 095

1205931= 04

50 08992 08994 08996 06888 06557100 09263 09261 09263 07870 07530150 09362 09366 09367 08176 07857200 09402 09398 09399 08350 08044

119886 = 095

1205931= minus05

50 09235 09239 09240 06378 06065100 09422 09420 09420 07342 07030150 09474 09479 09481 07820 07530200 09467 09471 09470 08057 07778

Acknowledgments

Lirsquos research is partially supported by the FundamentalResearch Funds for the Central Universities (no 2013XK03)Hersquos work is partially supported by the National NaturalScience Foundation of China (nos 11171230 and 11231010)

References

[1] TMikosch TGadrich C Kluppelberg andR J Adler ldquoParam-eter estimation for ARMA models with infinite variance inno-vationsrdquoTheAnnals of Statistics vol 23 no 1 pp 305ndash326 1995

[2] R A Davis ldquoGauss-Newton and M-estimation for ARMAprocesses with infinite variancerdquo Stochastic Processes and theirApplications vol 63 no 1 pp 75ndash95 1996

[3] J Pan HWang andQ Yao ldquoWeighted least absolute deviationsestimation forARMAmodels with infinite variancerdquoEconomet-ric Theory vol 23 no 5 pp 852ndash879 2007

[4] K Zhu and S Ling ldquoThe global LAD estimators for finiteinfinite variance ARMA(pq) modelsrdquo Econometric Theory vol28 no 5 pp 1065ndash1086 2012

[5] A B Owen ldquoEmpirical likelihood ratio confidence intervals fora single functionalrdquo Biometrika vol 75 no 2 pp 237ndash249 1988

[6] A BOwen ldquoEmpirical likelihood ratio confidence regionsrdquoTheAnnals of Statistics vol 18 no 1 pp 90ndash120 1990

10 Journal of Applied Mathematics

[7] A B Owen Empirical Likelihood Chapman and Hall LondonUK 2001

[8] A C Monti ldquoEmpirical likelihood confidence regions in timeseries modelsrdquo Biometrika vol 84 no 2 pp 395ndash405 1997

[9] C Chuang and N H Chan ldquoEmpirical likelihood for autore-gressive models with applications to unstable time seriesrdquoStatistica Sinica vol 12 no 2 pp 387ndash407 2002

[10] N H Chan L Peng and Y Qi ldquoQuantile inference for near-integrated autoregressive time series with infinite variancerdquoStatistica Sinica vol 16 no 1 pp 15ndash28 2006

[11] J Li W Liang S He and X Wu ldquoEmpirical likelihood forthe smoothed LAD estimator in infinite variance autoregressivemodelsrdquo Statistics amp Probability Letters vol 80 no 17-18 pp1420ndash1430 2010

[12] J Li W Liang and S He ldquoEmpirical likelihood for LADestimators in infinite variance ARMA modelsrdquo Statistics ampProbability Letters vol 81 no 2 pp 212ndash219 2011

[13] P J Brockwell and R A DavisTime seriesTheory andMethodsSpringer-Verlag New York NY USA 2nd edition 1991

[14] P C B Phillips ldquoA shortcut to LAD estimator asymptoticsrdquoEconometric Theory vol 7 no 4 pp 450ndash463 1991

[15] B W Silverman Density Estimation for Statistics and DataAnalysis Chapman ampHall London UK 1986

[16] V H de la Pena ldquoA general class of exponential inequalities formartingales and ratiosrdquoThe Annals of Probability vol 27 no 1pp 537ndash564 1999

[17] J Qin and J Lawless ldquoEmpirical likelihood and general estimat-ing equationsrdquo The Annals of Statistics vol 22 no 1 pp 300ndash325 1994

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

Page 4: Research Article Empirical Likelihood for Partial ...downloads.hindawi.com/journals/jam/2014/868970.pdf · Research Article Empirical Likelihood for Partial Parameters in ARMA Models

4 Journal of Applied Mathematics

differences with |119885119905119887119899119905| le 119862 where F

119905= 120590(120576

119904 119904 le 119905) For

some 1198620gt 0 by the ergodicity we have

1198812

119899=

119899

sum119905=119906+1

119864 (119885119905119887119899119905)2

| F119905minus1

=

119899

sum119905=119906+1

1198852

119905119864(119887119899119905)2

le 119862

119899

sum119905=119906+1

1198852

119905lt 119862 (119864 (119885

2

119905) + 1198620) 119899 as

(22)

Set119910 = 119862(119864(1198852119905)+1198620)119899 byTheorem 12A in [16] for all119860 gt 0

we have

119875

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum119905=119906+1

119885119905119887119899119905

1003816100381610038161003816100381610038161003816100381610038161003816

gt 119860radic119899 log 119899

= 119875

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum119905=119906+1

119885119905119887119899119905

1003816100381610038161003816100381610038161003816100381610038161003816

gt 119860radic119899 log 119899 1198812119899lt 119910 for some 119899

le 2 expminus1198602119899 log 119899

2 (119910 + 119862119860radic119899 log 119899)

= 2 expminus1198602 log 119899

2119862 (119864 (1198852119905) + 1198620) + 2119862119860radiclog 119899119899

(23)

Choosing 119860 such that 1198602 gt 2119862(119864(1198852119905) + 1198620) by the Borel-

Cantelli lemma the first term of (21) is119874(radiclog 119899119899) asThus1198701is 119874(radiclog 119899) as For 119870

2 by Davis [2] it holds that |120576

119905minus

120576119905(1206010)| le 120585119905 and 119860

119905(1206010)minus119876119905 le 120585119905 where 120585

119905= 119862sum

infin

119895=119905119903119895|119910119905minus119895|

for some 0 lt 119903 lt 1 Therefore

100381710038171003817100381711987021003817100381710038171003817 le

119862

ℎradic119899 minus 119906

119899

sum119905=119906+1

119908119905

100381710038171003817100381711987611990510038171003817100381710038171003816100381610038161003816120576119905 minus 120576119905 (1206010)

1003816100381610038161003816

+119862

radic119899 minus 119906

119899

sum119905=119906+1

119908119905

1003817100381710038171003817119876119905 minus 119860 119905 (1206010)1003817100381710038171003817

le119862

ℎradic119899 minus 119906

119899

sum119905=119906+1

120585119905+

119862

radic119899 minus 119906

119899

sum119905=119906+1

120585119905

=119862

ℎradic119899 minus 119906

119899

sum119905=119906+1

119903119905

infin

sum119897=0

119903119897 1003816100381610038161003816119910minus119897

1003816100381610038161003816

+119862

radic119899 minus 119906

119899

sum119905=119906+1

119903119905

infin

sum119897=0

119903119897 1003816100381610038161003816119910minus119897

1003816100381610038161003816as997888rarr 0

(24)

Thus1198702is 119874(radiclog 119899) as For119870

3 we have

100381710038171003817100381711987031003817100381710038171003817

le119862

radic119899 minus 119906

119899

sum119905=119906+1

1003816100381610038161003816119908119905 minus 11990811990510038161003816100381610038161003817100381710038171003817119860 119905 (1206010)

1003817100381710038171003817

le119862

radic119899 minus 119906

119899

sum119905=119906+1

suminfin

119896=119905119896minus120572

1003816100381610038161003816119910119905minus1198961003816100381610038161003816 sum119905minus1

119895=111990311989510038161003816100381610038161003816119910119905minus119895

10038161003816100381610038161003816

1 + sum119905minus1

119896=1119896minus120572

1003816100381610038161003816119910119905minus1198961003816100381610038161003816

le119862

radic119899 minus 119906

119899

sum119905=119906+1

infin

sum119896=119905

119896minus120572 1003816100381610038161003816119910119905minus119896

1003816100381610038161003816

119905minus1

sum119895=1

119903119895119895120572

le119862

radic119899 minus 119906

119899

sum119905=119906+1

infin

sum119896=119905

119896minus120572 1003816100381610038161003816119910119905minus119896

1003816100381610038161003816

le119862

radic119899 minus 119906

119899

sum119905=119906+1

infin

sum119897=0

(119905 + 119897)minus120572 1003816100381610038161003816119910minus119897

1003816100381610038161003816

le119862

radic119899 minus 119906

119899

sum119905=119906+1

(119905minus120572 10038161003816100381610038161199100

1003816100381610038161003816 +

infin

sum119897=1

2minus120572119905minus1205722

119897minus1205722 1003816100381610038161003816119910minus119897

1003816100381610038161003816)

=119862

radic119899 minus 119906

119899

sum119905=119906+1

119905minus120572 10038161003816100381610038161199100

1003816100381610038161003816

+119862

radic119899 minus 119906

119899

sum119905=119906+1

119905minus1205722

infin

sum119897=1

119897minus1205722 1003816100381610038161003816119910minus119897

1003816100381610038161003816as997888rarr 0

(25)

because we have the facts that 119860119905(1206010) le 119862sum

119905minus1

119895=1119903119895|119910119905minus119895| (see

[2]) and (119905+ 119897)minus120572 le 2minus120572(119905119897)minus1205722 for 119905 gt 0 119897 gt 0 Thus119870

3is also

119874(radiclog 119899) as Therefore part (i) holds For the proof of part(ii) we may write

120597119876119899ℎ(1206010)

120597120601120591

= minus1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119908119905119860119905(1206010) 119860120591

1119905(1206010)

times [119870(120576119905(1206010)

ℎ) + 119870(minus

120576119905(1206010)

ℎ)]

+1

119899 minus 119906

119899

sum119905=119906+1

119908119905

120597119860119905(1206010)

120597120601120591119866ℎ(120576119905(1206010))

= 1198631+ 1198632

(26)

where 1198601119905(120601) = minus120597120576

119905(120601)120597120601 For119863

1 we may write

1198631= minus

1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119908119905119876119905119876120591

1119905(119870(

120576119905

ℎ) + 119870(minus

120576119905

ℎ))

minus1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

[119908119905119860119905(1206010) 119860120591

1119905(1206010)

times (119870(120576119905(1206010)

ℎ) + 119870(minus

120576119905(1206010)

ℎ))

minus119908119905119876119905119876120591

1119905(119870(

120576119905

ℎ) + 119870(minus

120576119905

ℎ))]

= 11986311+ 11986312

(27)

Journal of Applied Mathematics 5

Note that

11986311= minus

1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119879119905119888119899119905minus

1

(119899 minus 119906) ℎ

times

119899

sum119905=119906+1

119908119905119876119905119876120591

1119905119864(119870(

120576119905

ℎ) + 119870(minus

120576119905

ℎ))

= minus1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119879119905119888119899119905

minus [1

119899 minus 119906

119899

sum119905=119906+1

119908119905119876119905119876120591

1119905] (2119891 (0) + 119900 (ℎ))

(28)

where 119879119905= 1199081199051198761199051198761205911119905and 119888119899119905= 119870(120576

119905ℎ) + 119870(minus120576

119905ℎ) minus 119864[119870(120576

119905

ℎ) + 119870(minus120576119905ℎ)] The second term of (28) is minus2119891(0)Σ

1as by

the ergodicity We will prove that the first term is 119900(1) asWe suppose that 119876

119905is the first element 119880

119905minus1without loss of

generality Note that for each 119899 ge 119906 + 1 119879119905119888119899119905F119905 119906 + 1 le

119905 le 119899 is a sequence ofmartingale differences with |119879119905119888119899119905| le 119862

and

1198812

119899=

119899

sum119905=119906+1

119864 (119879119905119888119899119905)2

| F119905minus1

=

119899

sum119905=119906+1

1198792

119905119864(119888119899119905)2

lt 119899119862 (119891 (0) 1198620ℎ + 119874 (ℎ

2)) as

(29)

where 1198620gt 0 is a constant Set 119910 = 119899119862(119891(0)119862

0ℎ + 119874(ℎ2)) by

Theorem 12A in [16] for all 120576 gt 0 we have

119875

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum119905=119906+1

119879119905119888119899119905

1003816100381610038161003816100381610038161003816100381610038161003816

gt (119899ℎ) 120576

= 119875

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum119905=119906+1

119879119905119888119899119905

1003816100381610038161003816100381610038161003816100381610038161003816

gt (119899ℎ) 120576 1198812

119899lt 119910 for some 119899

le 2 exp minus(119899ℎ)21205762

2 (119910 + 119862119899ℎ120576)

= 2 exp minus119899ℎ1205762

2119862 (119891 (0) 1198620+ 119874 (ℎ)) + 2119862120576

(30)

The result follows from the Borel-Cantelli lemmaThus11986311=

minus2119891(0)Σ1+ 119900(1) as Similar to119870

2and119870

3 we have

1003817100381710038171003817119863121003817100381710038171003817

le1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119908119905(1003817100381710038171003817119876119905 minus 119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198761119905

1003817100381710038171003817

+1003817100381710038171003817119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198761119905 minus 1198601119905 (1206010)

1003817100381710038171003817)

100381610038161003816100381610038161003816100381610038161003816119870(

120576119905(1206010)

ℎ) + 119870(minus

120576119905(1206010)

ℎ)

100381610038161003816100381610038161003816100381610038161003816

+1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

1003816100381610038161003816119908119905 minus 11990811990510038161003816100381610038161003817100381710038171003817119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198601119905 (1206010)

1003817100381710038171003817

times

100381610038161003816100381610038161003816100381610038161003816119870(

120576119905(1206010)

ℎ) + 119870(minus

120576119905(1206010)

ℎ)

100381610038161003816100381610038161003816100381610038161003816

+1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119908119905

1003817100381710038171003817119876119905100381710038171003817100381710038171003817100381710038171198761119905

1003817100381710038171003817

times

100381610038161003816100381610038161003816100381610038161003816[119870(

120576119905

ℎ) + 119870(minus

120576119905

ℎ)]

minus[119870(120576119905(1206010)

ℎ) + 119870(minus

120576119905(1206010)

ℎ)]

100381610038161003816100381610038161003816100381610038161003816

le119862

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119908119905(1003817100381710038171003817119876119905 minus 119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198761119905

1003817100381710038171003817

+1003817100381710038171003817119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198761119905 minus 1198601119905 (1206010)

1003817100381710038171003817)

+119862

(119899 minus 119906) ℎ

119899

sum119905=119906+1

1003816100381610038161003816119908119905 minus 11990811990510038161003816100381610038161003817100381710038171003817119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198601119905 (1206010)

1003817100381710038171003817

+119862

(119899 minus 119906) ℎ2

119899

sum119905=119906+1

119908119905

1003817100381710038171003817119876119905100381710038171003817100381710038171003817100381710038171198761119905

10038171003817100381710038171003816100381610038161003816120576119905 minus 120576119905 (1206010)

1003816100381610038161003816

le119862

(119899 minus 119906) ℎ

119899

sum119905=119906+1

120585119905+

119862

(119899 minus 119906) ℎ

119899

sum119905=119906+1

infin

sum119896=119905

119896minus120572 1003816100381610038161003816119910119905minus119896

1003816100381610038161003816

+119862

(119899 minus 119906) ℎ2

119899

sum119905=119906+1

120585119905

as997888rarr 0

(31)

Therefore 1198631

= minus2119891(0)Σ1+ 119900(1) as For 119863

2 from the

definition of 119860119905(120601) it holds for 119905 gt max(119901 119902) that

120579 (119861)120597119860119905119894(120601)

120597120601119895

= 0 119894 119895 = 1 119901

120579 (119861)120597119860119905119894(120601)

120597120601119895+119901

= minus119860119905minus119895119894

(120601) 119894 = 1 119901 119895 = 1 119902

120579 (119861)120597119860119905119895+119901

(120601)

120597120601119894

= minus119860119905minus119895119894

(120601) 119894 = 1 119901 119895 = 1 119902

120579 (119861)120597119860119905119894+119901

(120601)

120597120601119895+119901

= minus119860119905minus119895119894+119901

(120601) minus 119860119905minus119894119895+119901

(120601)

119894 119895 = 1 119902

(32)

6 Journal of Applied Mathematics

where 119861 is the backshift operator For 119905 = 0 plusmn1 plusmn2 define

1205790(119861)119883119905(119894119895)

= 0 119894 119895 = 1 119901

1205790(119861)119883119905(119894119895+119901)

= minus119876119905minus119895119894

119894 = 1 119901 119895 = 1 119902

1205790(119861)119883119905(119895+119901119894)

= minus119876119905minus119895119894

119894 = 1 119901 119895 = 1 119902

1205790(119861)119883119905(119894+119901119895+119901)

= minus119876119905minus119895119894+119901

minus 119876119905minus119894119895+119901

119894 119895 = 1 119902

(33)

where 119876119905119894

is the 119894th component of 119876119905 Put 119883

119905= (119883

119905(119894119895))

similar to [13] we have that 120597119860119905(1206010)120597120601120591 le 119862sum

119905minus1

119895=1119903119895|119910119905minus119895|

119883119905 le 119862sum

infin

119895=1119903119895|119910119905minus119895| and 119883

119905minus 120597119860119905(1206010)120597120601120591 le 120585

119905 Then

we may write

1198632=

1

119899 minus 119906

119899

sum119905=119906+1

119908119905119883119905119866ℎ(120576119905)

+1

119899 minus 119906

119899

sum119905=119906+1

[119908119905

120597119860119905(1206010)

120597120601120591119866ℎ(120576119905(1206010)) minus 119908

119905119883119905119866ℎ(120576119905)]

= 11986321+ 11986322

(34)

Similar to11986311and119863

12 we have that119863

21and119863

22are 119900(1) as

This completes the proof

Lemma 4 Under the conditions of Proposition 1 as 119899 rarr infin

(119894) 119885119899(120601) = 119900 (119899

13) as

(119894119894) 119878 (120601) = Ω + 119900 (1) as(35)

hold uniformly in 119861

Proof of Lemma 4 For part (i) from [2] we havethat 120597119860

119905(120601)120597120601120591 le 119862sum

119905minus1

119895=1119903119895|119910119905minus119895| and 119860

119905(120601) le

119862sum119905minus1

119895=1119903119895|119910119905minus119895| uniformly hold in the ball 119861 for sufficiently

large 119899 Then for each 120601 isin 119861 we have1003817100381710038171003817119898119905ℎ (120601)

1003817100381710038171003817 le 119908119905

1003817100381710038171003817119860 119905 (120601)1003817100381710038171003817

le119862sum119905minus1

119895=111990311989510038161003816100381610038161003816119910119905minus119895

10038161003816100381610038161003816

1 + sum119905minus1

119896=1119896minus120572

1003816100381610038161003816119910119905minus1198961003816100381610038161003816

le 119862

119905minus1

sum119896=1

119903119896119896120572

le 119862

infin

sum119896=1

119903119896119896120572lt infin

(36)

Thus part (i) holds For part (ii) similar to the proof ofLemma 3 we have that 119878(120601

0) = Ω + 119900(1) as For each 120601 isin 119861

by Taylor expansion we have

119878 (120601) minus 119878 (1206010)

=1

119899 minus 119906

119899

sum119905=119906+1

119898119905ℎ(1206010)

120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)

120591

+1

119899 minus 119906

119899

sum119905=119906+1

120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)119898120591

119905ℎ(1206010)

+1

119899 minus 119906

119899

sum119905=119906+1

120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)

times 120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)

120591

= 1198791+ 1198792+ 1198793

(37)

where 120601lowast lies between 1206010and 120601 For 119879

1 we have

100381710038171003817100381711987911003817100381710038171003817 le

1

119899 minus 119906

119899

sum119905=119906+1

1199082

119905

1003817100381710038171003817119860 119905 (1206010)1003817100381710038171003817

100381710038171003817100381710038171003817100381710038171003817

120597119860119905(120601lowast)

120597120601120591

100381710038171003817100381710038171003817100381710038171003817

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

+1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

1199082

119905

1003817100381710038171003817119860 119905 (1206010)10038171003817100381710038171003817100381710038171003817119860 119905 (120601

lowast)1003817100381710038171003817

times10038171003817100381710038171198601119905 (120601

lowast)10038171003817100381710038171003817100381710038171003817120601 minus 1206010

1003817100381710038171003817

times

100381610038161003816100381610038161003816100381610038161003816119870(

120576119905(120601lowast)

ℎ) + 119870(minus

120576119905(120601lowast)

ℎ)

100381610038161003816100381610038161003816100381610038161003816

le119862119889119899

119899 minus 119906

119899

sum119905=119906+1

(

119905minus1

sum119895=1

119903119895119895120572)

2

+119862119889119899

(119899 minus 119906) ℎ

119899

sum119905=119906+1

(

119905minus1

sum119895=1

119903119895119895120572)

3

as997888rarr 0

(38)

Similarly we have 1198792

as997888997888rarr 0 and 119879

3

as997888997888rarr 0 This completes the

proof

Proof of Proposition 1 For 120601 isin 119861 by Taylor expansion

119876119899ℎ(120601) = 119876

119899ℎ(1206010) +

120597119876119899ℎ(1206010)

120597120601120591(120601 minus 120601

0)

+1

2

119901+119902minus119898

sum119895119896=1

1205972119876119899ℎ(1206010)

120597120601119895120597120601119896

(120601119895minus 1206010

119895) (120601119896minus 1206010

119896)

+1

6

119901+119902minus119898

sum119895119896119897=1

1205973119876119899ℎ(120601lowast)

120597120601119895120597120601119896120597120601119897

times (120601119895minus 1206010

119895) (120601119896minus 1206010

119896) (120601119897minus 1206010

119897)

(39)

where 120601lowastlies between 120601

0and 120601 Note that the final term on

the right side of (39) can be written as

1

6

119901+119902minus119898

sum119895119896119897=1

[

[

(120601119895minus 1206010119895) (120601119896minus 1206010119896) (120601119897minus 1206010119897)

ℎ3]

]

times (ℎ3

119899 minus 119906

119899

sum119905=119906+1

1205973119898119905ℎ(120601lowast)

120597120601119895120597120601119896120597120601119897

)

(40)

Journal of Applied Mathematics 7

which is 119900(120575119899) as where 120575

119899= 120601 minus 120601

0 because 1198892

119899ℎ3 =

11198992120573minus3120574 rarr 0 and1003817100381710038171003817100381710038171003817100381710038171003817

ℎ3

119899 minus 119906

119899

sum119905=119906+1

1205973119898119905ℎ(120601lowast)

120597120601119895120597120601119896120597120601119897

1003817100381710038171003817100381710038171003817100381710038171003817

le 119862 (41)

The third term on the right side of (39) can be written as

1

2

119901+119902minus119898

sum119895119896=1

[

[

(120601119895minus 1206010

119895) (120601119896minus 1206010

119896)

ℎ]

]

times ℎ

119899 minus 119906

119899

sum119905=119906+1

1205972119898119905ℎ(1206010)

120597120601119895120597120601119896

(42)

which is also 119900(120575119899) as because 119889

119899ℎ = 1119899120573minus120574 rarr 0 and

ℎ

119899 minus 119906

119899

sum119905=119906+1

1205972119898119905ℎ(1206010)

120597120601119895120597120601119896

= 119900 (1) as (43)

by a similar proof of Lemma 3 Therefore

119876119899ℎ(120601) = 119876

119899ℎ(1206010) +

120597119876119899ℎ(1206010)

120597120601120591(120601 minus 120601

0) + 119900 (120575

119899) as

(44)

uniformly about 120601 isin 119861 Denote 120601 = 1206010+ 120583119889119899 for 120601 isin 120601

120601 minus 1206010 = 119889

119899 where 120583 = 1 Now we give a lower bound

for 119897ℎ(120601) on the surface of the ball Similar to [6] by Lemmas

3 and 4 we have

119897ℎ(120601)

= (119899 minus 119906)119876119899ℎ(120601)120591

119878(120601)minus1

119876119899ℎ(120601) + 119900 (119899

43minus3120573) as

= (119899 minus 119906)[

[

119874(radiclog 119899119899

)

+ (minus2119891 (0)) Σ1120583119889119899+ 119900 (119889

119899) ]

]

120591

Ωminus1

times [

[

119874(radiclog 119899119899

) + (minus2119891 (0)) Σ1120583119889119899+ 119900 (119889

119899)]

]

+ 119900 (11989943minus3120573

) as

ge (119888 minus 120576) 1198991minus2120573 as

(45)

where 119888 minus 120576 gt 0 and 119888 is the smallest eigenvalue of41198912(0)Σ120591

1Ωminus1Σ1 Similarly

119897ℎ(1206010)

= (119899 minus 119906)119876119899ℎ(1206010)120591

119878(1206010)minus1

119876119899ℎ(1206010) + 119900 (1) as

= 119874 (log 119899) as

(46)

Since 119897ℎ(120601) is a continuous function about 120601 as 120601 belongs to

the ball 119861 119897ℎ(120601) attains its minimum value at some point 120601

in the interior of this ball and 120601 satisfies 120597119897ℎ(120601)120597120601 = 0 it

follows that (12) holds This completes the proof

Proof of Theorem 2 Similar to the proof ofTheorem 2 of Qinand Lawless [17] we have

(

120601 minus 1206010

) = 119878minus1

119899(minus1198761119899(1206010 0) + 119900

119901(119899minus12)

119900119901(119899minus12

)) (47)

where

119878119899= (

1205971198761119899(1206010 0)

120597120582120591

1205971198761119899(1206010 0)

120597120601120591

1205971198762119899(1206010 0)

1205971205821205910

)119901

997888rarr (11987811

11987812

11987821

0)

= (minusΩ minus2119891 (0) Σ

1

minus2119891 (0) Σ120591

10

)

(48)

By the standard arguments in the proof of empirical likeli-hood (see [6]) we have

119897119901(1205960)

= minus (119899 minus 119906) (119876119899ℎ(1206010)120591

0) 119878minus1

119899(119876119899ℎ(1206010)120591

0)120591

+ 119900119901(1)

= (Ωminus12radic119899 minus 119906119876

119899ℎ(1206010))120591

times (119868 minus 41198912(0)Ωminus12

Σ1Δminus1Σ120591

1Ωminus12

)

times (Ωminus12radic119899 minus 119906119876

119899ℎ(1206010)) + 119900

119901(1)

(49)

where Δ = 41198912(0)Σ1205911Ωminus1Σ1 Sinceradic119899 minus 119906119876

119899ℎ(1206010)119889

997888rarr 119873(0Ω)

and

tr 41198912 (0)Ωminus12Σ1Δminus1Σ120591

1Ωminus12

= tr Δminus141198912 (0) Σ1205911Ωminus1Σ1 = 119901 + 119902 minus 119898

(50)

it follows that 119897119901(1205960)119889

997888rarr 1205942119898

4 Simulation Studies

We generated data from a simple ARMA(1 1) model 119910119905=

1205931119910119905minus1

+ 120576119905+ 1205991120576119905minus1

with119873(0 1) 1199052 and Cauchy innovation

distribution We set 119906 = 20 120572 = 3 and the true value(1205931 1205991) = (04 07) or (minus05 07) where 120593

1is the parameter

of interest The sample size 119899 = 50 100 150 200 and2000 replications are conducted in all cases We smooth theestimating equations using kernel

119870 (119909) =1

radic2120587120590119890minus119909221205902

(51)

8 Journal of Applied Mathematics

Table 1 The coverage probability of confidence intervals when 120576119905sim 119873(0 1)

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08818 08820 08822 08193 07875100 08898 08896 08897 08655 08431150 08926 08927 08932 08692 08395200 08983 08983 08986 08666 08363

119886 = 09

1205931= minus05

50 08888 08885 08892 08156 07813100 08967 08968 08973 08738 08448150 08943 08944 08946 08823 08574200 08972 08979 08977 08931 08692

119886 = 095

1205931= 04

50 09347 09350 09350 08724 08425100 09424 09430 09431 09123 08862150 09467 09471 09470 09157 08936200 09494 09494 09497 09160 08937

119886 = 095

1205931= minus05

50 09404 09404 09404 08705 08407100 09472 09474 09474 09134 08931150 09481 09479 09476 09248 09052200 09495 09495 09490 09326 09152

where 120590 = 01 which is the so-called Gaussian kernel Thecoverage probabilities of smoothed empirical likelihood con-fidence regions 119868

ℎ119888with the bandwidth ℎ = 1119899120574 are denot-

ed by EL(120574) where 120574 = 027 030 032 respectivelyAs another benchmark of the simulation experiments

we consider the confidence regions based on the asymptoticnormal distribution ofWLADEproposed by [3] To constructthe confidence regions we need to estimate 119891(0) Σ and ΩWe can estimate 119891(0) by

119891 (0) =1

119908119887119899(119899 minus 119906)

119899

sum119905=119906+1

119908119905

120576119905(120579)

119887119899

(52)

where (119909) = exp(minus119909)(1+exp(minus119909))2 is a kernel function on119877 and 119887

119899= 1119899] is a bandwidth

119908= (119899 minus 119906)

minus1sum119899

119905=119906+1119908119905 Σ

andΩ can be estimated respectively by

Σ =1

119899 minus 119906

119899

sum119905=119906+1

119908119905119876119905119876120591

119905

Ω =1

119899 minus 119906

119899

sum119905=119906+1

1199082

119905119876119905119876120591

119905

(53)

where 119876119905is defined in the same manner as 119876

119905 1205790is replaced

by 120579 and 120576119905is replaced by 120576

119905(120579) see (14) Based on this we can

construct a NA confidence region (ie based on the normalapproximation of WLADE) The coverage probabilities ofconfidence regions 119868NA based on the bandwidth 119887

119899= 1119899] are

denoted byNA(]) with ] = 025 020 respectively Tables 12

and 3 show the probabilities of the confidence intervals of 1205931

at confidence levels 09 and 095 respectivelyThe simulation results can be summarized as followsThe

coverage probabilities of NA(]) are much smaller than thenominal levels and very sensitive to the choice of bandwidth119887119899and 120576119905 On the other hand the coverage probabilities of

EL(120574) are much better and less sensitive to the choice ofbandwidth ℎ and 120576

119905 As the sample size 119899 increases the

coverage probabilities for both increase to the nominal levelsas one might expect

5 Conclusions

This paper explores a profile empirical likelihood method toconstruct confidence regions for the partial parameters ofinterest in IVARMAmodels We started with the foundationof estimating equations of WLADE then from there wederived smoothed empirical likelihood Moreover we haveproved that the resulting statistics has asymptotic standardchi-squared distribution Hence there is no need to estimateany additional quantity such as the asymptotic variance Thesimulations indeed show that the proposed method has agood finite sample behavior which experimentally confirmsour method

Conflict of Interests

The authors declare that they have no conflict of interests

Journal of Applied Mathematics 9

Table 2 The coverage probability of confidence intervals when 120576119905sim 1199052

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08774 08776 08779 07627 07290100 08850 08849 08856 08323 07995150 08919 08917 08924 08467 08157200 08935 08932 08932 08510 08229

119886 = 09

1205931= minus05

50 08902 08902 08899 07295 07055100 08942 08944 08950 08151 07836150 08946 08941 08937 08473 08204200 08965 08963 08961 08641 08392

119886 = 095

1205931= 04

50 09331 09327 09327 08163 07854100 09412 09412 09413 08834 08542150 09447 09446 09447 08965 08718200 09456 09458 09459 09001 08762

119886 = 095

1205931= minus05

50 09421 09418 09418 07963 07658100 09455 09462 09460 08695 08430150 09444 09442 09440 08910 08695200 09464 09463 09464 09072 08863

Table 3 The coverage probability of confidence intervals when 120576119905sim 119862119886119906119888ℎ119910

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08360 08358 08361 06345 06018100 08708 08708 08702 07286 06942150 08811 08819 08823 07614 07270200 08870 08864 08868 07800 07467

119886 = 09

1205931= minus05

50 08673 08679 08678 05818 05502100 08869 08870 08874 06756 06440150 08972 08974 08974 07232 06867200 08956 08955 08953 07510 07156

119886 = 095

1205931= 04

50 08992 08994 08996 06888 06557100 09263 09261 09263 07870 07530150 09362 09366 09367 08176 07857200 09402 09398 09399 08350 08044

119886 = 095

1205931= minus05

50 09235 09239 09240 06378 06065100 09422 09420 09420 07342 07030150 09474 09479 09481 07820 07530200 09467 09471 09470 08057 07778

Acknowledgments

Lirsquos research is partially supported by the FundamentalResearch Funds for the Central Universities (no 2013XK03)Hersquos work is partially supported by the National NaturalScience Foundation of China (nos 11171230 and 11231010)

References

[1] TMikosch TGadrich C Kluppelberg andR J Adler ldquoParam-eter estimation for ARMA models with infinite variance inno-vationsrdquoTheAnnals of Statistics vol 23 no 1 pp 305ndash326 1995

[2] R A Davis ldquoGauss-Newton and M-estimation for ARMAprocesses with infinite variancerdquo Stochastic Processes and theirApplications vol 63 no 1 pp 75ndash95 1996

[3] J Pan HWang andQ Yao ldquoWeighted least absolute deviationsestimation forARMAmodels with infinite variancerdquoEconomet-ric Theory vol 23 no 5 pp 852ndash879 2007

[4] K Zhu and S Ling ldquoThe global LAD estimators for finiteinfinite variance ARMA(pq) modelsrdquo Econometric Theory vol28 no 5 pp 1065ndash1086 2012

[5] A B Owen ldquoEmpirical likelihood ratio confidence intervals fora single functionalrdquo Biometrika vol 75 no 2 pp 237ndash249 1988

[6] A BOwen ldquoEmpirical likelihood ratio confidence regionsrdquoTheAnnals of Statistics vol 18 no 1 pp 90ndash120 1990

10 Journal of Applied Mathematics

[7] A B Owen Empirical Likelihood Chapman and Hall LondonUK 2001

[8] A C Monti ldquoEmpirical likelihood confidence regions in timeseries modelsrdquo Biometrika vol 84 no 2 pp 395ndash405 1997

[9] C Chuang and N H Chan ldquoEmpirical likelihood for autore-gressive models with applications to unstable time seriesrdquoStatistica Sinica vol 12 no 2 pp 387ndash407 2002

[10] N H Chan L Peng and Y Qi ldquoQuantile inference for near-integrated autoregressive time series with infinite variancerdquoStatistica Sinica vol 16 no 1 pp 15ndash28 2006

[11] J Li W Liang S He and X Wu ldquoEmpirical likelihood forthe smoothed LAD estimator in infinite variance autoregressivemodelsrdquo Statistics amp Probability Letters vol 80 no 17-18 pp1420ndash1430 2010

[12] J Li W Liang and S He ldquoEmpirical likelihood for LADestimators in infinite variance ARMA modelsrdquo Statistics ampProbability Letters vol 81 no 2 pp 212ndash219 2011

[13] P J Brockwell and R A DavisTime seriesTheory andMethodsSpringer-Verlag New York NY USA 2nd edition 1991

[14] P C B Phillips ldquoA shortcut to LAD estimator asymptoticsrdquoEconometric Theory vol 7 no 4 pp 450ndash463 1991

[15] B W Silverman Density Estimation for Statistics and DataAnalysis Chapman ampHall London UK 1986

[16] V H de la Pena ldquoA general class of exponential inequalities formartingales and ratiosrdquoThe Annals of Probability vol 27 no 1pp 537ndash564 1999

[17] J Qin and J Lawless ldquoEmpirical likelihood and general estimat-ing equationsrdquo The Annals of Statistics vol 22 no 1 pp 300ndash325 1994

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Research Article Empirical Likelihood for Partial ...downloads.hindawi.com/journals/jam/2014/868970.pdf · Research Article Empirical Likelihood for Partial Parameters in ARMA Models

Journal of Applied Mathematics 5

Note that

11986311= minus

1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119879119905119888119899119905minus

1

(119899 minus 119906) ℎ

times

119899

sum119905=119906+1

119908119905119876119905119876120591

1119905119864(119870(

120576119905

ℎ) + 119870(minus

120576119905

ℎ))

= minus1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119879119905119888119899119905

minus [1

119899 minus 119906

119899

sum119905=119906+1

119908119905119876119905119876120591

1119905] (2119891 (0) + 119900 (ℎ))

(28)

where 119879119905= 1199081199051198761199051198761205911119905and 119888119899119905= 119870(120576

119905ℎ) + 119870(minus120576

119905ℎ) minus 119864[119870(120576

119905

ℎ) + 119870(minus120576119905ℎ)] The second term of (28) is minus2119891(0)Σ

1as by

the ergodicity We will prove that the first term is 119900(1) asWe suppose that 119876

119905is the first element 119880

119905minus1without loss of

generality Note that for each 119899 ge 119906 + 1 119879119905119888119899119905F119905 119906 + 1 le

119905 le 119899 is a sequence ofmartingale differences with |119879119905119888119899119905| le 119862

and

1198812

119899=

119899

sum119905=119906+1

119864 (119879119905119888119899119905)2

| F119905minus1

=

119899

sum119905=119906+1

1198792

119905119864(119888119899119905)2

lt 119899119862 (119891 (0) 1198620ℎ + 119874 (ℎ

2)) as

(29)

where 1198620gt 0 is a constant Set 119910 = 119899119862(119891(0)119862

0ℎ + 119874(ℎ2)) by

Theorem 12A in [16] for all 120576 gt 0 we have

119875

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum119905=119906+1

119879119905119888119899119905

1003816100381610038161003816100381610038161003816100381610038161003816

gt (119899ℎ) 120576

= 119875

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum119905=119906+1

119879119905119888119899119905

1003816100381610038161003816100381610038161003816100381610038161003816

gt (119899ℎ) 120576 1198812

119899lt 119910 for some 119899

le 2 exp minus(119899ℎ)21205762

2 (119910 + 119862119899ℎ120576)

= 2 exp minus119899ℎ1205762

2119862 (119891 (0) 1198620+ 119874 (ℎ)) + 2119862120576

(30)

The result follows from the Borel-Cantelli lemmaThus11986311=

minus2119891(0)Σ1+ 119900(1) as Similar to119870

2and119870

3 we have

1003817100381710038171003817119863121003817100381710038171003817

le1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119908119905(1003817100381710038171003817119876119905 minus 119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198761119905

1003817100381710038171003817

+1003817100381710038171003817119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198761119905 minus 1198601119905 (1206010)

1003817100381710038171003817)

100381610038161003816100381610038161003816100381610038161003816119870(

120576119905(1206010)

ℎ) + 119870(minus

120576119905(1206010)

ℎ)

100381610038161003816100381610038161003816100381610038161003816

+1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

1003816100381610038161003816119908119905 minus 11990811990510038161003816100381610038161003817100381710038171003817119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198601119905 (1206010)

1003817100381710038171003817

times

100381610038161003816100381610038161003816100381610038161003816119870(

120576119905(1206010)

ℎ) + 119870(minus

120576119905(1206010)

ℎ)

100381610038161003816100381610038161003816100381610038161003816

+1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119908119905

1003817100381710038171003817119876119905100381710038171003817100381710038171003817100381710038171198761119905

1003817100381710038171003817

times

100381610038161003816100381610038161003816100381610038161003816[119870(

120576119905

ℎ) + 119870(minus

120576119905

ℎ)]

minus[119870(120576119905(1206010)

ℎ) + 119870(minus

120576119905(1206010)

ℎ)]

100381610038161003816100381610038161003816100381610038161003816

le119862

(119899 minus 119906) ℎ

119899

sum119905=119906+1

119908119905(1003817100381710038171003817119876119905 minus 119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198761119905

1003817100381710038171003817

+1003817100381710038171003817119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198761119905 minus 1198601119905 (1206010)

1003817100381710038171003817)

+119862

(119899 minus 119906) ℎ

119899

sum119905=119906+1

1003816100381610038161003816119908119905 minus 11990811990510038161003816100381610038161003817100381710038171003817119860 119905 (1206010)

100381710038171003817100381710038171003817100381710038171198601119905 (1206010)

1003817100381710038171003817

+119862

(119899 minus 119906) ℎ2

119899

sum119905=119906+1

119908119905

1003817100381710038171003817119876119905100381710038171003817100381710038171003817100381710038171198761119905

10038171003817100381710038171003816100381610038161003816120576119905 minus 120576119905 (1206010)

1003816100381610038161003816

le119862

(119899 minus 119906) ℎ

119899

sum119905=119906+1

120585119905+

119862

(119899 minus 119906) ℎ

119899

sum119905=119906+1

infin

sum119896=119905

119896minus120572 1003816100381610038161003816119910119905minus119896

1003816100381610038161003816

+119862

(119899 minus 119906) ℎ2

119899

sum119905=119906+1

120585119905

as997888rarr 0

(31)

Therefore 1198631

= minus2119891(0)Σ1+ 119900(1) as For 119863

2 from the

definition of 119860119905(120601) it holds for 119905 gt max(119901 119902) that

120579 (119861)120597119860119905119894(120601)

120597120601119895

= 0 119894 119895 = 1 119901

120579 (119861)120597119860119905119894(120601)

120597120601119895+119901

= minus119860119905minus119895119894

(120601) 119894 = 1 119901 119895 = 1 119902

120579 (119861)120597119860119905119895+119901

(120601)

120597120601119894

= minus119860119905minus119895119894

(120601) 119894 = 1 119901 119895 = 1 119902

120579 (119861)120597119860119905119894+119901

(120601)

120597120601119895+119901

= minus119860119905minus119895119894+119901

(120601) minus 119860119905minus119894119895+119901

(120601)

119894 119895 = 1 119902

(32)

6 Journal of Applied Mathematics

where 119861 is the backshift operator For 119905 = 0 plusmn1 plusmn2 define

1205790(119861)119883119905(119894119895)

= 0 119894 119895 = 1 119901

1205790(119861)119883119905(119894119895+119901)

= minus119876119905minus119895119894

119894 = 1 119901 119895 = 1 119902

1205790(119861)119883119905(119895+119901119894)

= minus119876119905minus119895119894

119894 = 1 119901 119895 = 1 119902

1205790(119861)119883119905(119894+119901119895+119901)

= minus119876119905minus119895119894+119901

minus 119876119905minus119894119895+119901

119894 119895 = 1 119902

(33)

where 119876119905119894

is the 119894th component of 119876119905 Put 119883

119905= (119883

119905(119894119895))

similar to [13] we have that 120597119860119905(1206010)120597120601120591 le 119862sum

119905minus1

119895=1119903119895|119910119905minus119895|

119883119905 le 119862sum

infin

119895=1119903119895|119910119905minus119895| and 119883

119905minus 120597119860119905(1206010)120597120601120591 le 120585

119905 Then

we may write

1198632=

1

119899 minus 119906

119899

sum119905=119906+1

119908119905119883119905119866ℎ(120576119905)

+1

119899 minus 119906

119899

sum119905=119906+1

[119908119905

120597119860119905(1206010)

120597120601120591119866ℎ(120576119905(1206010)) minus 119908

119905119883119905119866ℎ(120576119905)]

= 11986321+ 11986322

(34)

Similar to11986311and119863

12 we have that119863

21and119863

22are 119900(1) as

This completes the proof

Lemma 4 Under the conditions of Proposition 1 as 119899 rarr infin

(119894) 119885119899(120601) = 119900 (119899

13) as

(119894119894) 119878 (120601) = Ω + 119900 (1) as(35)

hold uniformly in 119861

Proof of Lemma 4 For part (i) from [2] we havethat 120597119860

119905(120601)120597120601120591 le 119862sum

119905minus1

119895=1119903119895|119910119905minus119895| and 119860

119905(120601) le

119862sum119905minus1

119895=1119903119895|119910119905minus119895| uniformly hold in the ball 119861 for sufficiently

large 119899 Then for each 120601 isin 119861 we have1003817100381710038171003817119898119905ℎ (120601)

1003817100381710038171003817 le 119908119905

1003817100381710038171003817119860 119905 (120601)1003817100381710038171003817

le119862sum119905minus1

119895=111990311989510038161003816100381610038161003816119910119905minus119895

10038161003816100381610038161003816

1 + sum119905minus1

119896=1119896minus120572

1003816100381610038161003816119910119905minus1198961003816100381610038161003816

le 119862

119905minus1

sum119896=1

119903119896119896120572

le 119862

infin

sum119896=1

119903119896119896120572lt infin

(36)

Thus part (i) holds For part (ii) similar to the proof ofLemma 3 we have that 119878(120601

0) = Ω + 119900(1) as For each 120601 isin 119861

by Taylor expansion we have

119878 (120601) minus 119878 (1206010)

=1

119899 minus 119906

119899

sum119905=119906+1

119898119905ℎ(1206010)

120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)

120591

+1

119899 minus 119906

119899

sum119905=119906+1

120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)119898120591

119905ℎ(1206010)

+1

119899 minus 119906

119899

sum119905=119906+1

120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)

times 120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)

120591

= 1198791+ 1198792+ 1198793

(37)

where 120601lowast lies between 1206010and 120601 For 119879

1 we have

100381710038171003817100381711987911003817100381710038171003817 le

1

119899 minus 119906

119899

sum119905=119906+1

1199082

119905

1003817100381710038171003817119860 119905 (1206010)1003817100381710038171003817

100381710038171003817100381710038171003817100381710038171003817

120597119860119905(120601lowast)

120597120601120591

100381710038171003817100381710038171003817100381710038171003817

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

+1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

1199082

119905

1003817100381710038171003817119860 119905 (1206010)10038171003817100381710038171003817100381710038171003817119860 119905 (120601

lowast)1003817100381710038171003817

times10038171003817100381710038171198601119905 (120601

lowast)10038171003817100381710038171003817100381710038171003817120601 minus 1206010

1003817100381710038171003817

times

100381610038161003816100381610038161003816100381610038161003816119870(

120576119905(120601lowast)

ℎ) + 119870(minus

120576119905(120601lowast)

ℎ)

100381610038161003816100381610038161003816100381610038161003816

le119862119889119899

119899 minus 119906

119899

sum119905=119906+1

(

119905minus1

sum119895=1

119903119895119895120572)

2

+119862119889119899

(119899 minus 119906) ℎ

119899

sum119905=119906+1

(

119905minus1

sum119895=1

119903119895119895120572)

3

as997888rarr 0

(38)

Similarly we have 1198792

as997888997888rarr 0 and 119879

3

as997888997888rarr 0 This completes the

proof

Proof of Proposition 1 For 120601 isin 119861 by Taylor expansion

119876119899ℎ(120601) = 119876

119899ℎ(1206010) +

120597119876119899ℎ(1206010)

120597120601120591(120601 minus 120601

0)

+1

2

119901+119902minus119898

sum119895119896=1

1205972119876119899ℎ(1206010)

120597120601119895120597120601119896

(120601119895minus 1206010

119895) (120601119896minus 1206010

119896)

+1

6

119901+119902minus119898

sum119895119896119897=1

1205973119876119899ℎ(120601lowast)

120597120601119895120597120601119896120597120601119897

times (120601119895minus 1206010

119895) (120601119896minus 1206010

119896) (120601119897minus 1206010

119897)

(39)

where 120601lowastlies between 120601

0and 120601 Note that the final term on

the right side of (39) can be written as

1

6

119901+119902minus119898

sum119895119896119897=1

[

[

(120601119895minus 1206010119895) (120601119896minus 1206010119896) (120601119897minus 1206010119897)

ℎ3]

]

times (ℎ3

119899 minus 119906

119899

sum119905=119906+1

1205973119898119905ℎ(120601lowast)

120597120601119895120597120601119896120597120601119897

)

(40)

Journal of Applied Mathematics 7

which is 119900(120575119899) as where 120575

119899= 120601 minus 120601

0 because 1198892

119899ℎ3 =

11198992120573minus3120574 rarr 0 and1003817100381710038171003817100381710038171003817100381710038171003817

ℎ3

119899 minus 119906

119899

sum119905=119906+1

1205973119898119905ℎ(120601lowast)

120597120601119895120597120601119896120597120601119897

1003817100381710038171003817100381710038171003817100381710038171003817

le 119862 (41)

The third term on the right side of (39) can be written as

1

2

119901+119902minus119898

sum119895119896=1

[

[

(120601119895minus 1206010

119895) (120601119896minus 1206010

119896)

ℎ]

]

times ℎ

119899 minus 119906

119899

sum119905=119906+1

1205972119898119905ℎ(1206010)

120597120601119895120597120601119896

(42)

which is also 119900(120575119899) as because 119889

119899ℎ = 1119899120573minus120574 rarr 0 and

ℎ

119899 minus 119906

119899

sum119905=119906+1

1205972119898119905ℎ(1206010)

120597120601119895120597120601119896

= 119900 (1) as (43)

by a similar proof of Lemma 3 Therefore

119876119899ℎ(120601) = 119876

119899ℎ(1206010) +

120597119876119899ℎ(1206010)

120597120601120591(120601 minus 120601

0) + 119900 (120575

119899) as

(44)

uniformly about 120601 isin 119861 Denote 120601 = 1206010+ 120583119889119899 for 120601 isin 120601

120601 minus 1206010 = 119889

119899 where 120583 = 1 Now we give a lower bound

for 119897ℎ(120601) on the surface of the ball Similar to [6] by Lemmas

3 and 4 we have

119897ℎ(120601)

= (119899 minus 119906)119876119899ℎ(120601)120591

119878(120601)minus1

119876119899ℎ(120601) + 119900 (119899

43minus3120573) as

= (119899 minus 119906)[

[

119874(radiclog 119899119899

)

+ (minus2119891 (0)) Σ1120583119889119899+ 119900 (119889

119899) ]

]

120591

Ωminus1

times [

[

119874(radiclog 119899119899

) + (minus2119891 (0)) Σ1120583119889119899+ 119900 (119889

119899)]

]

+ 119900 (11989943minus3120573

) as

ge (119888 minus 120576) 1198991minus2120573 as

(45)

where 119888 minus 120576 gt 0 and 119888 is the smallest eigenvalue of41198912(0)Σ120591

1Ωminus1Σ1 Similarly

119897ℎ(1206010)

= (119899 minus 119906)119876119899ℎ(1206010)120591

119878(1206010)minus1

119876119899ℎ(1206010) + 119900 (1) as

= 119874 (log 119899) as

(46)

Since 119897ℎ(120601) is a continuous function about 120601 as 120601 belongs to

the ball 119861 119897ℎ(120601) attains its minimum value at some point 120601

in the interior of this ball and 120601 satisfies 120597119897ℎ(120601)120597120601 = 0 it

follows that (12) holds This completes the proof

Proof of Theorem 2 Similar to the proof ofTheorem 2 of Qinand Lawless [17] we have

(

120601 minus 1206010

) = 119878minus1

119899(minus1198761119899(1206010 0) + 119900

119901(119899minus12)

119900119901(119899minus12

)) (47)

where

119878119899= (

1205971198761119899(1206010 0)

120597120582120591

1205971198761119899(1206010 0)

120597120601120591

1205971198762119899(1206010 0)

1205971205821205910

)119901

997888rarr (11987811

11987812

11987821

0)

= (minusΩ minus2119891 (0) Σ

1

minus2119891 (0) Σ120591

10

)

(48)

By the standard arguments in the proof of empirical likeli-hood (see [6]) we have

119897119901(1205960)

= minus (119899 minus 119906) (119876119899ℎ(1206010)120591

0) 119878minus1

119899(119876119899ℎ(1206010)120591

0)120591

+ 119900119901(1)

= (Ωminus12radic119899 minus 119906119876

119899ℎ(1206010))120591

times (119868 minus 41198912(0)Ωminus12

Σ1Δminus1Σ120591

1Ωminus12

)

times (Ωminus12radic119899 minus 119906119876

119899ℎ(1206010)) + 119900

119901(1)

(49)

where Δ = 41198912(0)Σ1205911Ωminus1Σ1 Sinceradic119899 minus 119906119876

119899ℎ(1206010)119889

997888rarr 119873(0Ω)

and

tr 41198912 (0)Ωminus12Σ1Δminus1Σ120591

1Ωminus12

= tr Δminus141198912 (0) Σ1205911Ωminus1Σ1 = 119901 + 119902 minus 119898

(50)

it follows that 119897119901(1205960)119889

997888rarr 1205942119898

4 Simulation Studies

We generated data from a simple ARMA(1 1) model 119910119905=

1205931119910119905minus1

+ 120576119905+ 1205991120576119905minus1

with119873(0 1) 1199052 and Cauchy innovation

distribution We set 119906 = 20 120572 = 3 and the true value(1205931 1205991) = (04 07) or (minus05 07) where 120593

1is the parameter

of interest The sample size 119899 = 50 100 150 200 and2000 replications are conducted in all cases We smooth theestimating equations using kernel

119870 (119909) =1

radic2120587120590119890minus119909221205902

(51)

8 Journal of Applied Mathematics

Table 1 The coverage probability of confidence intervals when 120576119905sim 119873(0 1)

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08818 08820 08822 08193 07875100 08898 08896 08897 08655 08431150 08926 08927 08932 08692 08395200 08983 08983 08986 08666 08363

119886 = 09

1205931= minus05

50 08888 08885 08892 08156 07813100 08967 08968 08973 08738 08448150 08943 08944 08946 08823 08574200 08972 08979 08977 08931 08692

119886 = 095

1205931= 04

50 09347 09350 09350 08724 08425100 09424 09430 09431 09123 08862150 09467 09471 09470 09157 08936200 09494 09494 09497 09160 08937

119886 = 095

1205931= minus05

50 09404 09404 09404 08705 08407100 09472 09474 09474 09134 08931150 09481 09479 09476 09248 09052200 09495 09495 09490 09326 09152

where 120590 = 01 which is the so-called Gaussian kernel Thecoverage probabilities of smoothed empirical likelihood con-fidence regions 119868

ℎ119888with the bandwidth ℎ = 1119899120574 are denot-

ed by EL(120574) where 120574 = 027 030 032 respectivelyAs another benchmark of the simulation experiments

we consider the confidence regions based on the asymptoticnormal distribution ofWLADEproposed by [3] To constructthe confidence regions we need to estimate 119891(0) Σ and ΩWe can estimate 119891(0) by

119891 (0) =1

119908119887119899(119899 minus 119906)

119899

sum119905=119906+1

119908119905

120576119905(120579)

119887119899

(52)

where (119909) = exp(minus119909)(1+exp(minus119909))2 is a kernel function on119877 and 119887

119899= 1119899] is a bandwidth

119908= (119899 minus 119906)

minus1sum119899

119905=119906+1119908119905 Σ

andΩ can be estimated respectively by

Σ =1

119899 minus 119906

119899

sum119905=119906+1

119908119905119876119905119876120591

119905

Ω =1

119899 minus 119906

119899

sum119905=119906+1

1199082

119905119876119905119876120591

119905

(53)

where 119876119905is defined in the same manner as 119876

119905 1205790is replaced

by 120579 and 120576119905is replaced by 120576

119905(120579) see (14) Based on this we can

construct a NA confidence region (ie based on the normalapproximation of WLADE) The coverage probabilities ofconfidence regions 119868NA based on the bandwidth 119887

119899= 1119899] are

denoted byNA(]) with ] = 025 020 respectively Tables 12

and 3 show the probabilities of the confidence intervals of 1205931

at confidence levels 09 and 095 respectivelyThe simulation results can be summarized as followsThe

coverage probabilities of NA(]) are much smaller than thenominal levels and very sensitive to the choice of bandwidth119887119899and 120576119905 On the other hand the coverage probabilities of

EL(120574) are much better and less sensitive to the choice ofbandwidth ℎ and 120576

119905 As the sample size 119899 increases the

coverage probabilities for both increase to the nominal levelsas one might expect

5 Conclusions

This paper explores a profile empirical likelihood method toconstruct confidence regions for the partial parameters ofinterest in IVARMAmodels We started with the foundationof estimating equations of WLADE then from there wederived smoothed empirical likelihood Moreover we haveproved that the resulting statistics has asymptotic standardchi-squared distribution Hence there is no need to estimateany additional quantity such as the asymptotic variance Thesimulations indeed show that the proposed method has agood finite sample behavior which experimentally confirmsour method

Conflict of Interests

The authors declare that they have no conflict of interests

Journal of Applied Mathematics 9

Table 2 The coverage probability of confidence intervals when 120576119905sim 1199052

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08774 08776 08779 07627 07290100 08850 08849 08856 08323 07995150 08919 08917 08924 08467 08157200 08935 08932 08932 08510 08229

119886 = 09

1205931= minus05

50 08902 08902 08899 07295 07055100 08942 08944 08950 08151 07836150 08946 08941 08937 08473 08204200 08965 08963 08961 08641 08392

119886 = 095

1205931= 04

50 09331 09327 09327 08163 07854100 09412 09412 09413 08834 08542150 09447 09446 09447 08965 08718200 09456 09458 09459 09001 08762

119886 = 095

1205931= minus05

50 09421 09418 09418 07963 07658100 09455 09462 09460 08695 08430150 09444 09442 09440 08910 08695200 09464 09463 09464 09072 08863

Table 3 The coverage probability of confidence intervals when 120576119905sim 119862119886119906119888ℎ119910

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08360 08358 08361 06345 06018100 08708 08708 08702 07286 06942150 08811 08819 08823 07614 07270200 08870 08864 08868 07800 07467

119886 = 09

1205931= minus05

50 08673 08679 08678 05818 05502100 08869 08870 08874 06756 06440150 08972 08974 08974 07232 06867200 08956 08955 08953 07510 07156

119886 = 095

1205931= 04

50 08992 08994 08996 06888 06557100 09263 09261 09263 07870 07530150 09362 09366 09367 08176 07857200 09402 09398 09399 08350 08044

119886 = 095

1205931= minus05

50 09235 09239 09240 06378 06065100 09422 09420 09420 07342 07030150 09474 09479 09481 07820 07530200 09467 09471 09470 08057 07778

Acknowledgments

Lirsquos research is partially supported by the FundamentalResearch Funds for the Central Universities (no 2013XK03)Hersquos work is partially supported by the National NaturalScience Foundation of China (nos 11171230 and 11231010)

References

[1] TMikosch TGadrich C Kluppelberg andR J Adler ldquoParam-eter estimation for ARMA models with infinite variance inno-vationsrdquoTheAnnals of Statistics vol 23 no 1 pp 305ndash326 1995

[2] R A Davis ldquoGauss-Newton and M-estimation for ARMAprocesses with infinite variancerdquo Stochastic Processes and theirApplications vol 63 no 1 pp 75ndash95 1996

[3] J Pan HWang andQ Yao ldquoWeighted least absolute deviationsestimation forARMAmodels with infinite variancerdquoEconomet-ric Theory vol 23 no 5 pp 852ndash879 2007

[4] K Zhu and S Ling ldquoThe global LAD estimators for finiteinfinite variance ARMA(pq) modelsrdquo Econometric Theory vol28 no 5 pp 1065ndash1086 2012

[5] A B Owen ldquoEmpirical likelihood ratio confidence intervals fora single functionalrdquo Biometrika vol 75 no 2 pp 237ndash249 1988

[6] A BOwen ldquoEmpirical likelihood ratio confidence regionsrdquoTheAnnals of Statistics vol 18 no 1 pp 90ndash120 1990

10 Journal of Applied Mathematics

[7] A B Owen Empirical Likelihood Chapman and Hall LondonUK 2001

[8] A C Monti ldquoEmpirical likelihood confidence regions in timeseries modelsrdquo Biometrika vol 84 no 2 pp 395ndash405 1997

[9] C Chuang and N H Chan ldquoEmpirical likelihood for autore-gressive models with applications to unstable time seriesrdquoStatistica Sinica vol 12 no 2 pp 387ndash407 2002

[10] N H Chan L Peng and Y Qi ldquoQuantile inference for near-integrated autoregressive time series with infinite variancerdquoStatistica Sinica vol 16 no 1 pp 15ndash28 2006

[11] J Li W Liang S He and X Wu ldquoEmpirical likelihood forthe smoothed LAD estimator in infinite variance autoregressivemodelsrdquo Statistics amp Probability Letters vol 80 no 17-18 pp1420ndash1430 2010

[12] J Li W Liang and S He ldquoEmpirical likelihood for LADestimators in infinite variance ARMA modelsrdquo Statistics ampProbability Letters vol 81 no 2 pp 212ndash219 2011

[13] P J Brockwell and R A DavisTime seriesTheory andMethodsSpringer-Verlag New York NY USA 2nd edition 1991

[14] P C B Phillips ldquoA shortcut to LAD estimator asymptoticsrdquoEconometric Theory vol 7 no 4 pp 450ndash463 1991

[15] B W Silverman Density Estimation for Statistics and DataAnalysis Chapman ampHall London UK 1986

[16] V H de la Pena ldquoA general class of exponential inequalities formartingales and ratiosrdquoThe Annals of Probability vol 27 no 1pp 537ndash564 1999

[17] J Qin and J Lawless ldquoEmpirical likelihood and general estimat-ing equationsrdquo The Annals of Statistics vol 22 no 1 pp 300ndash325 1994

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

Page 6: Research Article Empirical Likelihood for Partial ...downloads.hindawi.com/journals/jam/2014/868970.pdf · Research Article Empirical Likelihood for Partial Parameters in ARMA Models

6 Journal of Applied Mathematics

where 119861 is the backshift operator For 119905 = 0 plusmn1 plusmn2 define

1205790(119861)119883119905(119894119895)

= 0 119894 119895 = 1 119901

1205790(119861)119883119905(119894119895+119901)

= minus119876119905minus119895119894

119894 = 1 119901 119895 = 1 119902

1205790(119861)119883119905(119895+119901119894)

= minus119876119905minus119895119894

119894 = 1 119901 119895 = 1 119902

1205790(119861)119883119905(119894+119901119895+119901)

= minus119876119905minus119895119894+119901

minus 119876119905minus119894119895+119901

119894 119895 = 1 119902

(33)

where 119876119905119894

is the 119894th component of 119876119905 Put 119883

119905= (119883

119905(119894119895))

similar to [13] we have that 120597119860119905(1206010)120597120601120591 le 119862sum

119905minus1

119895=1119903119895|119910119905minus119895|

119883119905 le 119862sum

infin

119895=1119903119895|119910119905minus119895| and 119883

119905minus 120597119860119905(1206010)120597120601120591 le 120585

119905 Then

we may write

1198632=

1

119899 minus 119906

119899

sum119905=119906+1

119908119905119883119905119866ℎ(120576119905)

+1

119899 minus 119906

119899

sum119905=119906+1

[119908119905

120597119860119905(1206010)

120597120601120591119866ℎ(120576119905(1206010)) minus 119908

119905119883119905119866ℎ(120576119905)]

= 11986321+ 11986322

(34)

Similar to11986311and119863

12 we have that119863

21and119863

22are 119900(1) as

This completes the proof

Lemma 4 Under the conditions of Proposition 1 as 119899 rarr infin

(119894) 119885119899(120601) = 119900 (119899

13) as

(119894119894) 119878 (120601) = Ω + 119900 (1) as(35)

hold uniformly in 119861

Proof of Lemma 4 For part (i) from [2] we havethat 120597119860

119905(120601)120597120601120591 le 119862sum

119905minus1

119895=1119903119895|119910119905minus119895| and 119860

119905(120601) le

119862sum119905minus1

119895=1119903119895|119910119905minus119895| uniformly hold in the ball 119861 for sufficiently

large 119899 Then for each 120601 isin 119861 we have1003817100381710038171003817119898119905ℎ (120601)

1003817100381710038171003817 le 119908119905

1003817100381710038171003817119860 119905 (120601)1003817100381710038171003817

le119862sum119905minus1

119895=111990311989510038161003816100381610038161003816119910119905minus119895

10038161003816100381610038161003816

1 + sum119905minus1

119896=1119896minus120572

1003816100381610038161003816119910119905minus1198961003816100381610038161003816

le 119862

119905minus1

sum119896=1

119903119896119896120572

le 119862

infin

sum119896=1

119903119896119896120572lt infin

(36)

Thus part (i) holds For part (ii) similar to the proof ofLemma 3 we have that 119878(120601

0) = Ω + 119900(1) as For each 120601 isin 119861

by Taylor expansion we have

119878 (120601) minus 119878 (1206010)

=1

119899 minus 119906

119899

sum119905=119906+1

119898119905ℎ(1206010)

120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)

120591

+1

119899 minus 119906

119899

sum119905=119906+1

120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)119898120591

119905ℎ(1206010)

+1

119899 minus 119906

119899

sum119905=119906+1

120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)

times 120597119898119905ℎ(120601lowast)

120597120601120591(120601 minus 120601

0)

120591

= 1198791+ 1198792+ 1198793

(37)

where 120601lowast lies between 1206010and 120601 For 119879

1 we have

100381710038171003817100381711987911003817100381710038171003817 le

1

119899 minus 119906

119899

sum119905=119906+1

1199082

119905

1003817100381710038171003817119860 119905 (1206010)1003817100381710038171003817

100381710038171003817100381710038171003817100381710038171003817

120597119860119905(120601lowast)

120597120601120591

100381710038171003817100381710038171003817100381710038171003817

1003817100381710038171003817120601 minus 12060101003817100381710038171003817

+1

(119899 minus 119906) ℎ

119899

sum119905=119906+1

1199082

119905

1003817100381710038171003817119860 119905 (1206010)10038171003817100381710038171003817100381710038171003817119860 119905 (120601

lowast)1003817100381710038171003817

times10038171003817100381710038171198601119905 (120601

lowast)10038171003817100381710038171003817100381710038171003817120601 minus 1206010

1003817100381710038171003817

times

100381610038161003816100381610038161003816100381610038161003816119870(

120576119905(120601lowast)

ℎ) + 119870(minus

120576119905(120601lowast)

ℎ)

100381610038161003816100381610038161003816100381610038161003816

le119862119889119899

119899 minus 119906

119899

sum119905=119906+1

(

119905minus1

sum119895=1

119903119895119895120572)

2

+119862119889119899

(119899 minus 119906) ℎ

119899

sum119905=119906+1

(

119905minus1

sum119895=1

119903119895119895120572)

3

as997888rarr 0

(38)

Similarly we have 1198792

as997888997888rarr 0 and 119879

3

as997888997888rarr 0 This completes the

proof

Proof of Proposition 1 For 120601 isin 119861 by Taylor expansion

119876119899ℎ(120601) = 119876

119899ℎ(1206010) +

120597119876119899ℎ(1206010)

120597120601120591(120601 minus 120601

0)

+1

2

119901+119902minus119898

sum119895119896=1

1205972119876119899ℎ(1206010)

120597120601119895120597120601119896

(120601119895minus 1206010

119895) (120601119896minus 1206010

119896)

+1

6

119901+119902minus119898

sum119895119896119897=1

1205973119876119899ℎ(120601lowast)

120597120601119895120597120601119896120597120601119897

times (120601119895minus 1206010

119895) (120601119896minus 1206010

119896) (120601119897minus 1206010

119897)

(39)

where 120601lowastlies between 120601

0and 120601 Note that the final term on

the right side of (39) can be written as

1

6

119901+119902minus119898

sum119895119896119897=1

[

[

(120601119895minus 1206010119895) (120601119896minus 1206010119896) (120601119897minus 1206010119897)

ℎ3]

]

times (ℎ3

119899 minus 119906

119899

sum119905=119906+1

1205973119898119905ℎ(120601lowast)

120597120601119895120597120601119896120597120601119897

)

(40)

Journal of Applied Mathematics 7

which is 119900(120575119899) as where 120575

119899= 120601 minus 120601

0 because 1198892

119899ℎ3 =

11198992120573minus3120574 rarr 0 and1003817100381710038171003817100381710038171003817100381710038171003817

ℎ3

119899 minus 119906

119899

sum119905=119906+1

1205973119898119905ℎ(120601lowast)

120597120601119895120597120601119896120597120601119897

1003817100381710038171003817100381710038171003817100381710038171003817

le 119862 (41)

The third term on the right side of (39) can be written as

1

2

119901+119902minus119898

sum119895119896=1

[

[

(120601119895minus 1206010

119895) (120601119896minus 1206010

119896)

ℎ]

]

times ℎ

119899 minus 119906

119899

sum119905=119906+1

1205972119898119905ℎ(1206010)

120597120601119895120597120601119896

(42)

which is also 119900(120575119899) as because 119889

119899ℎ = 1119899120573minus120574 rarr 0 and

ℎ

119899 minus 119906

119899

sum119905=119906+1

1205972119898119905ℎ(1206010)

120597120601119895120597120601119896

= 119900 (1) as (43)

by a similar proof of Lemma 3 Therefore

119876119899ℎ(120601) = 119876

119899ℎ(1206010) +

120597119876119899ℎ(1206010)

120597120601120591(120601 minus 120601

0) + 119900 (120575

119899) as

(44)

uniformly about 120601 isin 119861 Denote 120601 = 1206010+ 120583119889119899 for 120601 isin 120601

120601 minus 1206010 = 119889

119899 where 120583 = 1 Now we give a lower bound

for 119897ℎ(120601) on the surface of the ball Similar to [6] by Lemmas

3 and 4 we have

119897ℎ(120601)

= (119899 minus 119906)119876119899ℎ(120601)120591

119878(120601)minus1

119876119899ℎ(120601) + 119900 (119899

43minus3120573) as

= (119899 minus 119906)[

[

119874(radiclog 119899119899

)

+ (minus2119891 (0)) Σ1120583119889119899+ 119900 (119889

119899) ]

]

120591

Ωminus1

times [

[

119874(radiclog 119899119899

) + (minus2119891 (0)) Σ1120583119889119899+ 119900 (119889

119899)]

]

+ 119900 (11989943minus3120573

) as

ge (119888 minus 120576) 1198991minus2120573 as

(45)

where 119888 minus 120576 gt 0 and 119888 is the smallest eigenvalue of41198912(0)Σ120591

1Ωminus1Σ1 Similarly

119897ℎ(1206010)

= (119899 minus 119906)119876119899ℎ(1206010)120591

119878(1206010)minus1

119876119899ℎ(1206010) + 119900 (1) as

= 119874 (log 119899) as

(46)

Since 119897ℎ(120601) is a continuous function about 120601 as 120601 belongs to

the ball 119861 119897ℎ(120601) attains its minimum value at some point 120601

in the interior of this ball and 120601 satisfies 120597119897ℎ(120601)120597120601 = 0 it

follows that (12) holds This completes the proof

Proof of Theorem 2 Similar to the proof ofTheorem 2 of Qinand Lawless [17] we have

(

120601 minus 1206010

) = 119878minus1

119899(minus1198761119899(1206010 0) + 119900

119901(119899minus12)

119900119901(119899minus12

)) (47)

where

119878119899= (

1205971198761119899(1206010 0)

120597120582120591

1205971198761119899(1206010 0)

120597120601120591

1205971198762119899(1206010 0)

1205971205821205910

)119901

997888rarr (11987811

11987812

11987821

0)

= (minusΩ minus2119891 (0) Σ

1

minus2119891 (0) Σ120591

10

)

(48)

By the standard arguments in the proof of empirical likeli-hood (see [6]) we have

119897119901(1205960)

= minus (119899 minus 119906) (119876119899ℎ(1206010)120591

0) 119878minus1

119899(119876119899ℎ(1206010)120591

0)120591

+ 119900119901(1)

= (Ωminus12radic119899 minus 119906119876

119899ℎ(1206010))120591

times (119868 minus 41198912(0)Ωminus12

Σ1Δminus1Σ120591

1Ωminus12

)

times (Ωminus12radic119899 minus 119906119876

119899ℎ(1206010)) + 119900

119901(1)

(49)

where Δ = 41198912(0)Σ1205911Ωminus1Σ1 Sinceradic119899 minus 119906119876

119899ℎ(1206010)119889

997888rarr 119873(0Ω)

and

tr 41198912 (0)Ωminus12Σ1Δminus1Σ120591

1Ωminus12

= tr Δminus141198912 (0) Σ1205911Ωminus1Σ1 = 119901 + 119902 minus 119898

(50)

it follows that 119897119901(1205960)119889

997888rarr 1205942119898

4 Simulation Studies

We generated data from a simple ARMA(1 1) model 119910119905=

1205931119910119905minus1

+ 120576119905+ 1205991120576119905minus1

with119873(0 1) 1199052 and Cauchy innovation

distribution We set 119906 = 20 120572 = 3 and the true value(1205931 1205991) = (04 07) or (minus05 07) where 120593

1is the parameter

of interest The sample size 119899 = 50 100 150 200 and2000 replications are conducted in all cases We smooth theestimating equations using kernel

119870 (119909) =1

radic2120587120590119890minus119909221205902

(51)

8 Journal of Applied Mathematics

Table 1 The coverage probability of confidence intervals when 120576119905sim 119873(0 1)

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08818 08820 08822 08193 07875100 08898 08896 08897 08655 08431150 08926 08927 08932 08692 08395200 08983 08983 08986 08666 08363

119886 = 09

1205931= minus05

50 08888 08885 08892 08156 07813100 08967 08968 08973 08738 08448150 08943 08944 08946 08823 08574200 08972 08979 08977 08931 08692

119886 = 095

1205931= 04

50 09347 09350 09350 08724 08425100 09424 09430 09431 09123 08862150 09467 09471 09470 09157 08936200 09494 09494 09497 09160 08937

119886 = 095

1205931= minus05

50 09404 09404 09404 08705 08407100 09472 09474 09474 09134 08931150 09481 09479 09476 09248 09052200 09495 09495 09490 09326 09152

where 120590 = 01 which is the so-called Gaussian kernel Thecoverage probabilities of smoothed empirical likelihood con-fidence regions 119868

ℎ119888with the bandwidth ℎ = 1119899120574 are denot-

ed by EL(120574) where 120574 = 027 030 032 respectivelyAs another benchmark of the simulation experiments

we consider the confidence regions based on the asymptoticnormal distribution ofWLADEproposed by [3] To constructthe confidence regions we need to estimate 119891(0) Σ and ΩWe can estimate 119891(0) by

119891 (0) =1

119908119887119899(119899 minus 119906)

119899

sum119905=119906+1

119908119905

120576119905(120579)

119887119899

(52)

where (119909) = exp(minus119909)(1+exp(minus119909))2 is a kernel function on119877 and 119887

119899= 1119899] is a bandwidth

119908= (119899 minus 119906)

minus1sum119899

119905=119906+1119908119905 Σ

andΩ can be estimated respectively by

Σ =1

119899 minus 119906

119899

sum119905=119906+1

119908119905119876119905119876120591

119905

Ω =1

119899 minus 119906

119899

sum119905=119906+1

1199082

119905119876119905119876120591

119905

(53)

where 119876119905is defined in the same manner as 119876

119905 1205790is replaced

by 120579 and 120576119905is replaced by 120576

119905(120579) see (14) Based on this we can

construct a NA confidence region (ie based on the normalapproximation of WLADE) The coverage probabilities ofconfidence regions 119868NA based on the bandwidth 119887

119899= 1119899] are

denoted byNA(]) with ] = 025 020 respectively Tables 12

and 3 show the probabilities of the confidence intervals of 1205931

at confidence levels 09 and 095 respectivelyThe simulation results can be summarized as followsThe

coverage probabilities of NA(]) are much smaller than thenominal levels and very sensitive to the choice of bandwidth119887119899and 120576119905 On the other hand the coverage probabilities of

EL(120574) are much better and less sensitive to the choice ofbandwidth ℎ and 120576

119905 As the sample size 119899 increases the

coverage probabilities for both increase to the nominal levelsas one might expect

5 Conclusions

This paper explores a profile empirical likelihood method toconstruct confidence regions for the partial parameters ofinterest in IVARMAmodels We started with the foundationof estimating equations of WLADE then from there wederived smoothed empirical likelihood Moreover we haveproved that the resulting statistics has asymptotic standardchi-squared distribution Hence there is no need to estimateany additional quantity such as the asymptotic variance Thesimulations indeed show that the proposed method has agood finite sample behavior which experimentally confirmsour method

Conflict of Interests

The authors declare that they have no conflict of interests

Journal of Applied Mathematics 9

Table 2 The coverage probability of confidence intervals when 120576119905sim 1199052

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08774 08776 08779 07627 07290100 08850 08849 08856 08323 07995150 08919 08917 08924 08467 08157200 08935 08932 08932 08510 08229

119886 = 09

1205931= minus05

50 08902 08902 08899 07295 07055100 08942 08944 08950 08151 07836150 08946 08941 08937 08473 08204200 08965 08963 08961 08641 08392

119886 = 095

1205931= 04

50 09331 09327 09327 08163 07854100 09412 09412 09413 08834 08542150 09447 09446 09447 08965 08718200 09456 09458 09459 09001 08762

119886 = 095

1205931= minus05

50 09421 09418 09418 07963 07658100 09455 09462 09460 08695 08430150 09444 09442 09440 08910 08695200 09464 09463 09464 09072 08863

Table 3 The coverage probability of confidence intervals when 120576119905sim 119862119886119906119888ℎ119910

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08360 08358 08361 06345 06018100 08708 08708 08702 07286 06942150 08811 08819 08823 07614 07270200 08870 08864 08868 07800 07467

119886 = 09

1205931= minus05

50 08673 08679 08678 05818 05502100 08869 08870 08874 06756 06440150 08972 08974 08974 07232 06867200 08956 08955 08953 07510 07156

119886 = 095

1205931= 04

50 08992 08994 08996 06888 06557100 09263 09261 09263 07870 07530150 09362 09366 09367 08176 07857200 09402 09398 09399 08350 08044

119886 = 095

1205931= minus05

50 09235 09239 09240 06378 06065100 09422 09420 09420 07342 07030150 09474 09479 09481 07820 07530200 09467 09471 09470 08057 07778

Acknowledgments

Lirsquos research is partially supported by the FundamentalResearch Funds for the Central Universities (no 2013XK03)Hersquos work is partially supported by the National NaturalScience Foundation of China (nos 11171230 and 11231010)

References

[1] TMikosch TGadrich C Kluppelberg andR J Adler ldquoParam-eter estimation for ARMA models with infinite variance inno-vationsrdquoTheAnnals of Statistics vol 23 no 1 pp 305ndash326 1995

[2] R A Davis ldquoGauss-Newton and M-estimation for ARMAprocesses with infinite variancerdquo Stochastic Processes and theirApplications vol 63 no 1 pp 75ndash95 1996

[3] J Pan HWang andQ Yao ldquoWeighted least absolute deviationsestimation forARMAmodels with infinite variancerdquoEconomet-ric Theory vol 23 no 5 pp 852ndash879 2007

[4] K Zhu and S Ling ldquoThe global LAD estimators for finiteinfinite variance ARMA(pq) modelsrdquo Econometric Theory vol28 no 5 pp 1065ndash1086 2012

[5] A B Owen ldquoEmpirical likelihood ratio confidence intervals fora single functionalrdquo Biometrika vol 75 no 2 pp 237ndash249 1988

[6] A BOwen ldquoEmpirical likelihood ratio confidence regionsrdquoTheAnnals of Statistics vol 18 no 1 pp 90ndash120 1990

10 Journal of Applied Mathematics

[7] A B Owen Empirical Likelihood Chapman and Hall LondonUK 2001

[8] A C Monti ldquoEmpirical likelihood confidence regions in timeseries modelsrdquo Biometrika vol 84 no 2 pp 395ndash405 1997

[9] C Chuang and N H Chan ldquoEmpirical likelihood for autore-gressive models with applications to unstable time seriesrdquoStatistica Sinica vol 12 no 2 pp 387ndash407 2002

[10] N H Chan L Peng and Y Qi ldquoQuantile inference for near-integrated autoregressive time series with infinite variancerdquoStatistica Sinica vol 16 no 1 pp 15ndash28 2006

[11] J Li W Liang S He and X Wu ldquoEmpirical likelihood forthe smoothed LAD estimator in infinite variance autoregressivemodelsrdquo Statistics amp Probability Letters vol 80 no 17-18 pp1420ndash1430 2010

[12] J Li W Liang and S He ldquoEmpirical likelihood for LADestimators in infinite variance ARMA modelsrdquo Statistics ampProbability Letters vol 81 no 2 pp 212ndash219 2011

[13] P J Brockwell and R A DavisTime seriesTheory andMethodsSpringer-Verlag New York NY USA 2nd edition 1991

[14] P C B Phillips ldquoA shortcut to LAD estimator asymptoticsrdquoEconometric Theory vol 7 no 4 pp 450ndash463 1991

[15] B W Silverman Density Estimation for Statistics and DataAnalysis Chapman ampHall London UK 1986

[16] V H de la Pena ldquoA general class of exponential inequalities formartingales and ratiosrdquoThe Annals of Probability vol 27 no 1pp 537ndash564 1999

[17] J Qin and J Lawless ldquoEmpirical likelihood and general estimat-ing equationsrdquo The Annals of Statistics vol 22 no 1 pp 300ndash325 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article Empirical Likelihood for Partial ...downloads.hindawi.com/journals/jam/2014/868970.pdf · Research Article Empirical Likelihood for Partial Parameters in ARMA Models

Journal of Applied Mathematics 7

which is 119900(120575119899) as where 120575

119899= 120601 minus 120601

0 because 1198892

119899ℎ3 =

11198992120573minus3120574 rarr 0 and1003817100381710038171003817100381710038171003817100381710038171003817

ℎ3

119899 minus 119906

119899

sum119905=119906+1

1205973119898119905ℎ(120601lowast)

120597120601119895120597120601119896120597120601119897

1003817100381710038171003817100381710038171003817100381710038171003817

le 119862 (41)

The third term on the right side of (39) can be written as

1

2

119901+119902minus119898

sum119895119896=1

[

[

(120601119895minus 1206010

119895) (120601119896minus 1206010

119896)

ℎ]

]

times ℎ

119899 minus 119906

119899

sum119905=119906+1

1205972119898119905ℎ(1206010)

120597120601119895120597120601119896

(42)

which is also 119900(120575119899) as because 119889

119899ℎ = 1119899120573minus120574 rarr 0 and

ℎ

119899 minus 119906

119899

sum119905=119906+1

1205972119898119905ℎ(1206010)

120597120601119895120597120601119896

= 119900 (1) as (43)

by a similar proof of Lemma 3 Therefore

119876119899ℎ(120601) = 119876

119899ℎ(1206010) +

120597119876119899ℎ(1206010)

120597120601120591(120601 minus 120601

0) + 119900 (120575

119899) as

(44)

uniformly about 120601 isin 119861 Denote 120601 = 1206010+ 120583119889119899 for 120601 isin 120601

120601 minus 1206010 = 119889

119899 where 120583 = 1 Now we give a lower bound

for 119897ℎ(120601) on the surface of the ball Similar to [6] by Lemmas

3 and 4 we have

119897ℎ(120601)

= (119899 minus 119906)119876119899ℎ(120601)120591

119878(120601)minus1

119876119899ℎ(120601) + 119900 (119899

43minus3120573) as

= (119899 minus 119906)[

[

119874(radiclog 119899119899

)

+ (minus2119891 (0)) Σ1120583119889119899+ 119900 (119889

119899) ]

]

120591

Ωminus1

times [

[

119874(radiclog 119899119899

) + (minus2119891 (0)) Σ1120583119889119899+ 119900 (119889

119899)]

]

+ 119900 (11989943minus3120573

) as

ge (119888 minus 120576) 1198991minus2120573 as

(45)

where 119888 minus 120576 gt 0 and 119888 is the smallest eigenvalue of41198912(0)Σ120591

1Ωminus1Σ1 Similarly

119897ℎ(1206010)

= (119899 minus 119906)119876119899ℎ(1206010)120591

119878(1206010)minus1

119876119899ℎ(1206010) + 119900 (1) as

= 119874 (log 119899) as

(46)

Since 119897ℎ(120601) is a continuous function about 120601 as 120601 belongs to

the ball 119861 119897ℎ(120601) attains its minimum value at some point 120601

in the interior of this ball and 120601 satisfies 120597119897ℎ(120601)120597120601 = 0 it

follows that (12) holds This completes the proof

Proof of Theorem 2 Similar to the proof ofTheorem 2 of Qinand Lawless [17] we have

(

120601 minus 1206010

) = 119878minus1

119899(minus1198761119899(1206010 0) + 119900

119901(119899minus12)

119900119901(119899minus12

)) (47)

where

119878119899= (

1205971198761119899(1206010 0)

120597120582120591

1205971198761119899(1206010 0)

120597120601120591

1205971198762119899(1206010 0)

1205971205821205910

)119901

997888rarr (11987811

11987812

11987821

0)

= (minusΩ minus2119891 (0) Σ

1

minus2119891 (0) Σ120591

10

)

(48)

By the standard arguments in the proof of empirical likeli-hood (see [6]) we have

119897119901(1205960)

= minus (119899 minus 119906) (119876119899ℎ(1206010)120591

0) 119878minus1

119899(119876119899ℎ(1206010)120591

0)120591

+ 119900119901(1)

= (Ωminus12radic119899 minus 119906119876

119899ℎ(1206010))120591

times (119868 minus 41198912(0)Ωminus12

Σ1Δminus1Σ120591

1Ωminus12

)

times (Ωminus12radic119899 minus 119906119876

119899ℎ(1206010)) + 119900

119901(1)

(49)

where Δ = 41198912(0)Σ1205911Ωminus1Σ1 Sinceradic119899 minus 119906119876

119899ℎ(1206010)119889

997888rarr 119873(0Ω)

and

tr 41198912 (0)Ωminus12Σ1Δminus1Σ120591

1Ωminus12

= tr Δminus141198912 (0) Σ1205911Ωminus1Σ1 = 119901 + 119902 minus 119898

(50)

it follows that 119897119901(1205960)119889

997888rarr 1205942119898

4 Simulation Studies

We generated data from a simple ARMA(1 1) model 119910119905=

1205931119910119905minus1

+ 120576119905+ 1205991120576119905minus1

with119873(0 1) 1199052 and Cauchy innovation

distribution We set 119906 = 20 120572 = 3 and the true value(1205931 1205991) = (04 07) or (minus05 07) where 120593

1is the parameter

of interest The sample size 119899 = 50 100 150 200 and2000 replications are conducted in all cases We smooth theestimating equations using kernel

119870 (119909) =1

radic2120587120590119890minus119909221205902

(51)

8 Journal of Applied Mathematics

Table 1 The coverage probability of confidence intervals when 120576119905sim 119873(0 1)

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08818 08820 08822 08193 07875100 08898 08896 08897 08655 08431150 08926 08927 08932 08692 08395200 08983 08983 08986 08666 08363

119886 = 09

1205931= minus05

50 08888 08885 08892 08156 07813100 08967 08968 08973 08738 08448150 08943 08944 08946 08823 08574200 08972 08979 08977 08931 08692

119886 = 095

1205931= 04

50 09347 09350 09350 08724 08425100 09424 09430 09431 09123 08862150 09467 09471 09470 09157 08936200 09494 09494 09497 09160 08937

119886 = 095

1205931= minus05

50 09404 09404 09404 08705 08407100 09472 09474 09474 09134 08931150 09481 09479 09476 09248 09052200 09495 09495 09490 09326 09152

where 120590 = 01 which is the so-called Gaussian kernel Thecoverage probabilities of smoothed empirical likelihood con-fidence regions 119868

ℎ119888with the bandwidth ℎ = 1119899120574 are denot-

ed by EL(120574) where 120574 = 027 030 032 respectivelyAs another benchmark of the simulation experiments

we consider the confidence regions based on the asymptoticnormal distribution ofWLADEproposed by [3] To constructthe confidence regions we need to estimate 119891(0) Σ and ΩWe can estimate 119891(0) by

119891 (0) =1

119908119887119899(119899 minus 119906)

119899

sum119905=119906+1

119908119905

120576119905(120579)

119887119899

(52)

where (119909) = exp(minus119909)(1+exp(minus119909))2 is a kernel function on119877 and 119887

119899= 1119899] is a bandwidth

119908= (119899 minus 119906)

minus1sum119899

119905=119906+1119908119905 Σ

andΩ can be estimated respectively by

Σ =1

119899 minus 119906

119899

sum119905=119906+1

119908119905119876119905119876120591

119905

Ω =1

119899 minus 119906

119899

sum119905=119906+1

1199082

119905119876119905119876120591

119905

(53)

where 119876119905is defined in the same manner as 119876

119905 1205790is replaced

by 120579 and 120576119905is replaced by 120576

119905(120579) see (14) Based on this we can

construct a NA confidence region (ie based on the normalapproximation of WLADE) The coverage probabilities ofconfidence regions 119868NA based on the bandwidth 119887

119899= 1119899] are

denoted byNA(]) with ] = 025 020 respectively Tables 12

and 3 show the probabilities of the confidence intervals of 1205931

at confidence levels 09 and 095 respectivelyThe simulation results can be summarized as followsThe

coverage probabilities of NA(]) are much smaller than thenominal levels and very sensitive to the choice of bandwidth119887119899and 120576119905 On the other hand the coverage probabilities of

EL(120574) are much better and less sensitive to the choice ofbandwidth ℎ and 120576

119905 As the sample size 119899 increases the

coverage probabilities for both increase to the nominal levelsas one might expect

5 Conclusions

This paper explores a profile empirical likelihood method toconstruct confidence regions for the partial parameters ofinterest in IVARMAmodels We started with the foundationof estimating equations of WLADE then from there wederived smoothed empirical likelihood Moreover we haveproved that the resulting statistics has asymptotic standardchi-squared distribution Hence there is no need to estimateany additional quantity such as the asymptotic variance Thesimulations indeed show that the proposed method has agood finite sample behavior which experimentally confirmsour method

Conflict of Interests

The authors declare that they have no conflict of interests

Journal of Applied Mathematics 9

Table 2 The coverage probability of confidence intervals when 120576119905sim 1199052

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08774 08776 08779 07627 07290100 08850 08849 08856 08323 07995150 08919 08917 08924 08467 08157200 08935 08932 08932 08510 08229

119886 = 09

1205931= minus05

50 08902 08902 08899 07295 07055100 08942 08944 08950 08151 07836150 08946 08941 08937 08473 08204200 08965 08963 08961 08641 08392

119886 = 095

1205931= 04

50 09331 09327 09327 08163 07854100 09412 09412 09413 08834 08542150 09447 09446 09447 08965 08718200 09456 09458 09459 09001 08762

119886 = 095

1205931= minus05

50 09421 09418 09418 07963 07658100 09455 09462 09460 08695 08430150 09444 09442 09440 08910 08695200 09464 09463 09464 09072 08863

Table 3 The coverage probability of confidence intervals when 120576119905sim 119862119886119906119888ℎ119910

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08360 08358 08361 06345 06018100 08708 08708 08702 07286 06942150 08811 08819 08823 07614 07270200 08870 08864 08868 07800 07467

119886 = 09

1205931= minus05

50 08673 08679 08678 05818 05502100 08869 08870 08874 06756 06440150 08972 08974 08974 07232 06867200 08956 08955 08953 07510 07156

119886 = 095

1205931= 04

50 08992 08994 08996 06888 06557100 09263 09261 09263 07870 07530150 09362 09366 09367 08176 07857200 09402 09398 09399 08350 08044

119886 = 095

1205931= minus05

50 09235 09239 09240 06378 06065100 09422 09420 09420 07342 07030150 09474 09479 09481 07820 07530200 09467 09471 09470 08057 07778

Acknowledgments

Lirsquos research is partially supported by the FundamentalResearch Funds for the Central Universities (no 2013XK03)Hersquos work is partially supported by the National NaturalScience Foundation of China (nos 11171230 and 11231010)

References

[1] TMikosch TGadrich C Kluppelberg andR J Adler ldquoParam-eter estimation for ARMA models with infinite variance inno-vationsrdquoTheAnnals of Statistics vol 23 no 1 pp 305ndash326 1995

[2] R A Davis ldquoGauss-Newton and M-estimation for ARMAprocesses with infinite variancerdquo Stochastic Processes and theirApplications vol 63 no 1 pp 75ndash95 1996

[3] J Pan HWang andQ Yao ldquoWeighted least absolute deviationsestimation forARMAmodels with infinite variancerdquoEconomet-ric Theory vol 23 no 5 pp 852ndash879 2007

[4] K Zhu and S Ling ldquoThe global LAD estimators for finiteinfinite variance ARMA(pq) modelsrdquo Econometric Theory vol28 no 5 pp 1065ndash1086 2012

[5] A B Owen ldquoEmpirical likelihood ratio confidence intervals fora single functionalrdquo Biometrika vol 75 no 2 pp 237ndash249 1988

[6] A BOwen ldquoEmpirical likelihood ratio confidence regionsrdquoTheAnnals of Statistics vol 18 no 1 pp 90ndash120 1990

10 Journal of Applied Mathematics

[7] A B Owen Empirical Likelihood Chapman and Hall LondonUK 2001

[8] A C Monti ldquoEmpirical likelihood confidence regions in timeseries modelsrdquo Biometrika vol 84 no 2 pp 395ndash405 1997

[9] C Chuang and N H Chan ldquoEmpirical likelihood for autore-gressive models with applications to unstable time seriesrdquoStatistica Sinica vol 12 no 2 pp 387ndash407 2002

[10] N H Chan L Peng and Y Qi ldquoQuantile inference for near-integrated autoregressive time series with infinite variancerdquoStatistica Sinica vol 16 no 1 pp 15ndash28 2006

[11] J Li W Liang S He and X Wu ldquoEmpirical likelihood forthe smoothed LAD estimator in infinite variance autoregressivemodelsrdquo Statistics amp Probability Letters vol 80 no 17-18 pp1420ndash1430 2010

[12] J Li W Liang and S He ldquoEmpirical likelihood for LADestimators in infinite variance ARMA modelsrdquo Statistics ampProbability Letters vol 81 no 2 pp 212ndash219 2011

[13] P J Brockwell and R A DavisTime seriesTheory andMethodsSpringer-Verlag New York NY USA 2nd edition 1991

[14] P C B Phillips ldquoA shortcut to LAD estimator asymptoticsrdquoEconometric Theory vol 7 no 4 pp 450ndash463 1991

[15] B W Silverman Density Estimation for Statistics and DataAnalysis Chapman ampHall London UK 1986

[16] V H de la Pena ldquoA general class of exponential inequalities formartingales and ratiosrdquoThe Annals of Probability vol 27 no 1pp 537ndash564 1999

[17] J Qin and J Lawless ldquoEmpirical likelihood and general estimat-ing equationsrdquo The Annals of Statistics vol 22 no 1 pp 300ndash325 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Research Article Empirical Likelihood for Partial ...downloads.hindawi.com/journals/jam/2014/868970.pdf · Research Article Empirical Likelihood for Partial Parameters in ARMA Models

8 Journal of Applied Mathematics

Table 1 The coverage probability of confidence intervals when 120576119905sim 119873(0 1)

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08818 08820 08822 08193 07875100 08898 08896 08897 08655 08431150 08926 08927 08932 08692 08395200 08983 08983 08986 08666 08363

119886 = 09

1205931= minus05

50 08888 08885 08892 08156 07813100 08967 08968 08973 08738 08448150 08943 08944 08946 08823 08574200 08972 08979 08977 08931 08692

119886 = 095

1205931= 04

50 09347 09350 09350 08724 08425100 09424 09430 09431 09123 08862150 09467 09471 09470 09157 08936200 09494 09494 09497 09160 08937

119886 = 095

1205931= minus05

50 09404 09404 09404 08705 08407100 09472 09474 09474 09134 08931150 09481 09479 09476 09248 09052200 09495 09495 09490 09326 09152

where 120590 = 01 which is the so-called Gaussian kernel Thecoverage probabilities of smoothed empirical likelihood con-fidence regions 119868

ℎ119888with the bandwidth ℎ = 1119899120574 are denot-

ed by EL(120574) where 120574 = 027 030 032 respectivelyAs another benchmark of the simulation experiments

we consider the confidence regions based on the asymptoticnormal distribution ofWLADEproposed by [3] To constructthe confidence regions we need to estimate 119891(0) Σ and ΩWe can estimate 119891(0) by

119891 (0) =1

119908119887119899(119899 minus 119906)

119899

sum119905=119906+1

119908119905

120576119905(120579)

119887119899

(52)

where (119909) = exp(minus119909)(1+exp(minus119909))2 is a kernel function on119877 and 119887

119899= 1119899] is a bandwidth

119908= (119899 minus 119906)

minus1sum119899

119905=119906+1119908119905 Σ

andΩ can be estimated respectively by

Σ =1

119899 minus 119906

119899

sum119905=119906+1

119908119905119876119905119876120591

119905

Ω =1

119899 minus 119906

119899

sum119905=119906+1

1199082

119905119876119905119876120591

119905

(53)

where 119876119905is defined in the same manner as 119876

119905 1205790is replaced

by 120579 and 120576119905is replaced by 120576

119905(120579) see (14) Based on this we can

construct a NA confidence region (ie based on the normalapproximation of WLADE) The coverage probabilities ofconfidence regions 119868NA based on the bandwidth 119887

119899= 1119899] are

denoted byNA(]) with ] = 025 020 respectively Tables 12

and 3 show the probabilities of the confidence intervals of 1205931

at confidence levels 09 and 095 respectivelyThe simulation results can be summarized as followsThe

coverage probabilities of NA(]) are much smaller than thenominal levels and very sensitive to the choice of bandwidth119887119899and 120576119905 On the other hand the coverage probabilities of

EL(120574) are much better and less sensitive to the choice ofbandwidth ℎ and 120576

119905 As the sample size 119899 increases the

coverage probabilities for both increase to the nominal levelsas one might expect

5 Conclusions

This paper explores a profile empirical likelihood method toconstruct confidence regions for the partial parameters ofinterest in IVARMAmodels We started with the foundationof estimating equations of WLADE then from there wederived smoothed empirical likelihood Moreover we haveproved that the resulting statistics has asymptotic standardchi-squared distribution Hence there is no need to estimateany additional quantity such as the asymptotic variance Thesimulations indeed show that the proposed method has agood finite sample behavior which experimentally confirmsour method

Conflict of Interests

The authors declare that they have no conflict of interests

Journal of Applied Mathematics 9

Table 2 The coverage probability of confidence intervals when 120576119905sim 1199052

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08774 08776 08779 07627 07290100 08850 08849 08856 08323 07995150 08919 08917 08924 08467 08157200 08935 08932 08932 08510 08229

119886 = 09

1205931= minus05

50 08902 08902 08899 07295 07055100 08942 08944 08950 08151 07836150 08946 08941 08937 08473 08204200 08965 08963 08961 08641 08392

119886 = 095

1205931= 04

50 09331 09327 09327 08163 07854100 09412 09412 09413 08834 08542150 09447 09446 09447 08965 08718200 09456 09458 09459 09001 08762

119886 = 095

1205931= minus05

50 09421 09418 09418 07963 07658100 09455 09462 09460 08695 08430150 09444 09442 09440 08910 08695200 09464 09463 09464 09072 08863

Table 3 The coverage probability of confidence intervals when 120576119905sim 119862119886119906119888ℎ119910

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08360 08358 08361 06345 06018100 08708 08708 08702 07286 06942150 08811 08819 08823 07614 07270200 08870 08864 08868 07800 07467

119886 = 09

1205931= minus05

50 08673 08679 08678 05818 05502100 08869 08870 08874 06756 06440150 08972 08974 08974 07232 06867200 08956 08955 08953 07510 07156

119886 = 095

1205931= 04

50 08992 08994 08996 06888 06557100 09263 09261 09263 07870 07530150 09362 09366 09367 08176 07857200 09402 09398 09399 08350 08044

119886 = 095

1205931= minus05

50 09235 09239 09240 06378 06065100 09422 09420 09420 07342 07030150 09474 09479 09481 07820 07530200 09467 09471 09470 08057 07778

Acknowledgments

Lirsquos research is partially supported by the FundamentalResearch Funds for the Central Universities (no 2013XK03)Hersquos work is partially supported by the National NaturalScience Foundation of China (nos 11171230 and 11231010)

References

[1] TMikosch TGadrich C Kluppelberg andR J Adler ldquoParam-eter estimation for ARMA models with infinite variance inno-vationsrdquoTheAnnals of Statistics vol 23 no 1 pp 305ndash326 1995

[2] R A Davis ldquoGauss-Newton and M-estimation for ARMAprocesses with infinite variancerdquo Stochastic Processes and theirApplications vol 63 no 1 pp 75ndash95 1996

[3] J Pan HWang andQ Yao ldquoWeighted least absolute deviationsestimation forARMAmodels with infinite variancerdquoEconomet-ric Theory vol 23 no 5 pp 852ndash879 2007

[4] K Zhu and S Ling ldquoThe global LAD estimators for finiteinfinite variance ARMA(pq) modelsrdquo Econometric Theory vol28 no 5 pp 1065ndash1086 2012

[5] A B Owen ldquoEmpirical likelihood ratio confidence intervals fora single functionalrdquo Biometrika vol 75 no 2 pp 237ndash249 1988

[6] A BOwen ldquoEmpirical likelihood ratio confidence regionsrdquoTheAnnals of Statistics vol 18 no 1 pp 90ndash120 1990

10 Journal of Applied Mathematics

[7] A B Owen Empirical Likelihood Chapman and Hall LondonUK 2001

[8] A C Monti ldquoEmpirical likelihood confidence regions in timeseries modelsrdquo Biometrika vol 84 no 2 pp 395ndash405 1997

[9] C Chuang and N H Chan ldquoEmpirical likelihood for autore-gressive models with applications to unstable time seriesrdquoStatistica Sinica vol 12 no 2 pp 387ndash407 2002

[10] N H Chan L Peng and Y Qi ldquoQuantile inference for near-integrated autoregressive time series with infinite variancerdquoStatistica Sinica vol 16 no 1 pp 15ndash28 2006

[11] J Li W Liang S He and X Wu ldquoEmpirical likelihood forthe smoothed LAD estimator in infinite variance autoregressivemodelsrdquo Statistics amp Probability Letters vol 80 no 17-18 pp1420ndash1430 2010

[12] J Li W Liang and S He ldquoEmpirical likelihood for LADestimators in infinite variance ARMA modelsrdquo Statistics ampProbability Letters vol 81 no 2 pp 212ndash219 2011

[13] P J Brockwell and R A DavisTime seriesTheory andMethodsSpringer-Verlag New York NY USA 2nd edition 1991

[14] P C B Phillips ldquoA shortcut to LAD estimator asymptoticsrdquoEconometric Theory vol 7 no 4 pp 450ndash463 1991

[15] B W Silverman Density Estimation for Statistics and DataAnalysis Chapman ampHall London UK 1986

[16] V H de la Pena ldquoA general class of exponential inequalities formartingales and ratiosrdquoThe Annals of Probability vol 27 no 1pp 537ndash564 1999

[17] J Qin and J Lawless ldquoEmpirical likelihood and general estimat-ing equationsrdquo The Annals of Statistics vol 22 no 1 pp 300ndash325 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Research Article Empirical Likelihood for Partial ...downloads.hindawi.com/journals/jam/2014/868970.pdf · Research Article Empirical Likelihood for Partial Parameters in ARMA Models

Journal of Applied Mathematics 9

Table 2 The coverage probability of confidence intervals when 120576119905sim 1199052

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08774 08776 08779 07627 07290100 08850 08849 08856 08323 07995150 08919 08917 08924 08467 08157200 08935 08932 08932 08510 08229

119886 = 09

1205931= minus05

50 08902 08902 08899 07295 07055100 08942 08944 08950 08151 07836150 08946 08941 08937 08473 08204200 08965 08963 08961 08641 08392

119886 = 095

1205931= 04

50 09331 09327 09327 08163 07854100 09412 09412 09413 08834 08542150 09447 09446 09447 08965 08718200 09456 09458 09459 09001 08762

119886 = 095

1205931= minus05

50 09421 09418 09418 07963 07658100 09455 09462 09460 08695 08430150 09444 09442 09440 08910 08695200 09464 09463 09464 09072 08863

Table 3 The coverage probability of confidence intervals when 120576119905sim 119862119886119906119888ℎ119910

119899 EL(027) EL(030) EL(032) NA(025) NA(020)

119886 = 09

1205931= 04

50 08360 08358 08361 06345 06018100 08708 08708 08702 07286 06942150 08811 08819 08823 07614 07270200 08870 08864 08868 07800 07467

119886 = 09

1205931= minus05

50 08673 08679 08678 05818 05502100 08869 08870 08874 06756 06440150 08972 08974 08974 07232 06867200 08956 08955 08953 07510 07156

119886 = 095

1205931= 04

50 08992 08994 08996 06888 06557100 09263 09261 09263 07870 07530150 09362 09366 09367 08176 07857200 09402 09398 09399 08350 08044

119886 = 095

1205931= minus05

50 09235 09239 09240 06378 06065100 09422 09420 09420 07342 07030150 09474 09479 09481 07820 07530200 09467 09471 09470 08057 07778

Acknowledgments

Lirsquos research is partially supported by the FundamentalResearch Funds for the Central Universities (no 2013XK03)Hersquos work is partially supported by the National NaturalScience Foundation of China (nos 11171230 and 11231010)

References

[1] TMikosch TGadrich C Kluppelberg andR J Adler ldquoParam-eter estimation for ARMA models with infinite variance inno-vationsrdquoTheAnnals of Statistics vol 23 no 1 pp 305ndash326 1995

[2] R A Davis ldquoGauss-Newton and M-estimation for ARMAprocesses with infinite variancerdquo Stochastic Processes and theirApplications vol 63 no 1 pp 75ndash95 1996

[3] J Pan HWang andQ Yao ldquoWeighted least absolute deviationsestimation forARMAmodels with infinite variancerdquoEconomet-ric Theory vol 23 no 5 pp 852ndash879 2007

[4] K Zhu and S Ling ldquoThe global LAD estimators for finiteinfinite variance ARMA(pq) modelsrdquo Econometric Theory vol28 no 5 pp 1065ndash1086 2012

[5] A B Owen ldquoEmpirical likelihood ratio confidence intervals fora single functionalrdquo Biometrika vol 75 no 2 pp 237ndash249 1988

[6] A BOwen ldquoEmpirical likelihood ratio confidence regionsrdquoTheAnnals of Statistics vol 18 no 1 pp 90ndash120 1990

10 Journal of Applied Mathematics

[7] A B Owen Empirical Likelihood Chapman and Hall LondonUK 2001

[8] A C Monti ldquoEmpirical likelihood confidence regions in timeseries modelsrdquo Biometrika vol 84 no 2 pp 395ndash405 1997

[9] C Chuang and N H Chan ldquoEmpirical likelihood for autore-gressive models with applications to unstable time seriesrdquoStatistica Sinica vol 12 no 2 pp 387ndash407 2002

[10] N H Chan L Peng and Y Qi ldquoQuantile inference for near-integrated autoregressive time series with infinite variancerdquoStatistica Sinica vol 16 no 1 pp 15ndash28 2006

[11] J Li W Liang S He and X Wu ldquoEmpirical likelihood forthe smoothed LAD estimator in infinite variance autoregressivemodelsrdquo Statistics amp Probability Letters vol 80 no 17-18 pp1420ndash1430 2010

[12] J Li W Liang and S He ldquoEmpirical likelihood for LADestimators in infinite variance ARMA modelsrdquo Statistics ampProbability Letters vol 81 no 2 pp 212ndash219 2011

[13] P J Brockwell and R A DavisTime seriesTheory andMethodsSpringer-Verlag New York NY USA 2nd edition 1991

[14] P C B Phillips ldquoA shortcut to LAD estimator asymptoticsrdquoEconometric Theory vol 7 no 4 pp 450ndash463 1991

[15] B W Silverman Density Estimation for Statistics and DataAnalysis Chapman ampHall London UK 1986

[16] V H de la Pena ldquoA general class of exponential inequalities formartingales and ratiosrdquoThe Annals of Probability vol 27 no 1pp 537ndash564 1999

[17] J Qin and J Lawless ldquoEmpirical likelihood and general estimat-ing equationsrdquo The Annals of Statistics vol 22 no 1 pp 300ndash325 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Research Article Empirical Likelihood for Partial ...downloads.hindawi.com/journals/jam/2014/868970.pdf · Research Article Empirical Likelihood for Partial Parameters in ARMA Models

10 Journal of Applied Mathematics

[7] A B Owen Empirical Likelihood Chapman and Hall LondonUK 2001

[8] A C Monti ldquoEmpirical likelihood confidence regions in timeseries modelsrdquo Biometrika vol 84 no 2 pp 395ndash405 1997

[9] C Chuang and N H Chan ldquoEmpirical likelihood for autore-gressive models with applications to unstable time seriesrdquoStatistica Sinica vol 12 no 2 pp 387ndash407 2002

[10] N H Chan L Peng and Y Qi ldquoQuantile inference for near-integrated autoregressive time series with infinite variancerdquoStatistica Sinica vol 16 no 1 pp 15ndash28 2006

[11] J Li W Liang S He and X Wu ldquoEmpirical likelihood forthe smoothed LAD estimator in infinite variance autoregressivemodelsrdquo Statistics amp Probability Letters vol 80 no 17-18 pp1420ndash1430 2010

[12] J Li W Liang and S He ldquoEmpirical likelihood for LADestimators in infinite variance ARMA modelsrdquo Statistics ampProbability Letters vol 81 no 2 pp 212ndash219 2011

[13] P J Brockwell and R A DavisTime seriesTheory andMethodsSpringer-Verlag New York NY USA 2nd edition 1991

[14] P C B Phillips ldquoA shortcut to LAD estimator asymptoticsrdquoEconometric Theory vol 7 no 4 pp 450ndash463 1991

[15] B W Silverman Density Estimation for Statistics and DataAnalysis Chapman ampHall London UK 1986

[16] V H de la Pena ldquoA general class of exponential inequalities formartingales and ratiosrdquoThe Annals of Probability vol 27 no 1pp 537ndash564 1999

[17] J Qin and J Lawless ldquoEmpirical likelihood and general estimat-ing equationsrdquo The Annals of Statistics vol 22 no 1 pp 300ndash325 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 11: Research Article Empirical Likelihood for Partial ...downloads.hindawi.com/journals/jam/2014/868970.pdf · Research Article Empirical Likelihood for Partial Parameters in ARMA Models

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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Jackknife Empirical Likelihood Inference For The … EMPIRICAL LIKELIHOOD INFERENCE FOR THE PIETRA RATIO by YUEJU SU Under the Direction of Yichuan Zhao, PhD ABSTRACT Pietra ratio

Jackknife Empirical Likelihood Inference For The … EMPIRICAL LIKELIHOOD INFERENCE FOR THE PIETRA RATIO by YUEJU SU Under the Direction of Yichuan Zhao, PhD ABSTRACT Pietra ratio

Empirical Likelihood Estimation for Population ...

Empirical Likelihood Estimation for Population ...

Local polynomial fitting based on empirical likelihood fileLocal polynomial fitting based on empirical likelihood Citation for published version (APA): Zhang, J., & Liu, A. (2000).

Local polynomial fitting based on empirical likelihood fileLocal polynomial fitting based on empirical likelihood Citation for published version (APA): Zhang, J., & Liu, A. (2000).

Partial Maximum Likelihood Estimation of Spatial Probit Models · 2010-10-18 · Partial Maximum Likelihood Estimation of Spatial Probit Models ∗ Honglin Wang† Michigan State

Partial Maximum Likelihood Estimation of Spatial Probit Models · 2010-10-18 · Partial Maximum Likelihood Estimation of Spatial Probit Models ∗ Honglin Wang† Michigan State

Nonparametric empirical Bayes and maximum likelihood ... · Nonparametric empirical Bayes and maximum likelihood estimation for high-dimensional ... and Mizera’s work, ...

Nonparametric empirical Bayes and maximum likelihood ... · Nonparametric empirical Bayes and maximum likelihood estimation for high-dimensional ... and Mizera’s work, ...

Sieve-based Empirical Likelihood under Semiparametric ...

Sieve-based Empirical Likelihood under Semiparametric ...

AN EMPIRICAL LIKELIHOOD RATIO TEST FOR NORMALITY

AN EMPIRICAL LIKELIHOOD RATIO TEST FOR NORMALITY

ABC and empirical likelihood

ABC and empirical likelihood

A Simple Empirical Likelihood Ratio Test for ... - Hindawi

A Simple Empirical Likelihood Ratio Test for ... - Hindawi

The VIII Conference Survey Sampling ... - web.ue.katowice.pl · Modelling survey data using empirical likelihood Yves Berger We propose a new design-based empirical likelihood approach

The VIII Conference Survey Sampling ... - web.ue.katowice.pl · Modelling survey data using empirical likelihood Yves Berger We propose a new design-based empirical likelihood approach

Generalized Empirical Likelihood Inference in Partially ...

Generalized Empirical Likelihood Inference in Partially ...

Shrinkage empirical likelihood estimator in …...Biometrics 000, 000{000 DOI: 000 000 0000 Shrinkage empirical likelihood estimator in longitudinal analysis with time-dependent covariates{

Shrinkage empirical likelihood estimator in …...Biometrics 000, 000{000 DOI: 000 000 0000 Shrinkage empirical likelihood estimator in longitudinal analysis with time-dependent covariates{

Empirical Likelihood Estimation for Population ......generalized linear models, including general least square GLS method, quasi-likelihood method, normal theory maximum likelihood,

Empirical Likelihood Estimation for Population ......generalized linear models, including general least square GLS method, quasi-likelihood method, normal theory maximum likelihood,

On the Second Order Properties of Empirical Likelihood ...meg/MEG2004/Chen-Song-Xi.pdf · On the Second Order Properties of Empirical Likelihood with Moment Restrictions Song Xi Chena

On the Second Order Properties of Empirical Likelihood ...meg/MEG2004/Chen-Song-Xi.pdf · On the Second Order Properties of Empirical Likelihood with Moment Restrictions Song Xi Chena

Sieve Empirical Likelihood and Extensions of the ... · Sieve Empirical Likelihood and Extensions ... of the above parametric likelihood, arise naturally. One ... the parametric likelihood

Sieve Empirical Likelihood and Extensions of the ... · Sieve Empirical Likelihood and Extensions ... of the above parametric likelihood, arise naturally. One ... the parametric likelihood

Computing Bayesian posterior with empirical likelihood in population genetics

Computing Bayesian posterior with empirical likelihood in population genetics

Blockwise empirical likelihood for time series of …...Blockwise empirical likelihood for time series of counts Rongning Wua,∗, Jiguo Caob aDepartment of Statistics and Computer

Blockwise empirical likelihood for time series of …...Blockwise empirical likelihood for time series of counts Rongning Wua,∗, Jiguo Caob aDepartment of Statistics and Computer

Calibration of the Empirical Likelihood Method for a Vector …statweb.stanford.edu/~owen/reports/EL_Calibration.pdfIn this paper, we focus on the use of the empirical likelihood method

Calibration of the Empirical Likelihood Method for a Vector …statweb.stanford.edu/~owen/reports/EL_Calibration.pdfIn this paper, we focus on the use of the empirical likelihood method

Diagnostic measures for empirical likelihood of general ...

Diagnostic measures for empirical likelihood of general ...

Jackknife empirical likelihood method for copulasyqi/mydownload/12test.pdfJackknife empirical likelihood method for copulas 75 sciences. Although some commonly used dependence measures

Jackknife empirical likelihood method for copulasyqi/mydownload/12test.pdfJackknife empirical likelihood method for copulas 75 sciences. Although some commonly used dependence measures