Research Article Testing Heteroscedasticity in...

Research ArticleTesting Heteroscedasticity in Nonparametric Regression Basedon Trend Analysis

Si-Lian Shen1 Jian-Ling Cui2 and Chun-Wei Wang1

1 School of Mathematics and Statistics Henan University of Science and Technology Luoyang 471003 China2 Electronic Equipment Test Center Luoyang 471003 China

Correspondence should be addressed to Si-Lian Shen meipersuit126com

Received 27 January 2014 Revised 26 May 2014 Accepted 26 May 2014 Published 11 June 2014

Academic Editor Zhihua Zhang

Copyright copy 2014 Si-Lian Shen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We first propose in this paper a new test method for detecting heteroscedasticity of the error term in nonparametric regressionSome simulation experiments are then conducted to evaluate the performance of the proposed methodology A real-world data setis finally analyzed to demonstrate the application of the method

1 Introduction

In recent years nonparametric regression models have beenwidely applied in a variety of areas for data analysis Theestimation of the regression function and related statisticalinferences in nonparametric models are usually based on theassumption that the error term is homoscedastic However inmany real-world problems we rarely know a priori whetherthis assumption can be guaranteed Therefore it is necessaryto develop a method for detecting heteroscedasticity in theerror terms before we embark on the model fitting andinferential issues

In the literature of the statistical nonparametric regres-sion there have been many papers on testing heteroscedas-ticity (see eg [1ndash8]) Among these papers a procedure wasdeveloped by Dette and Munk [2] based on an estimatorfor the best 119871

2-approximation of the variance function bya constant and was extended by You and Chen [5] to par-tially linear regression models Dette [1] proposed a test forheteroscedasticity in nonparametric regression A residual-based statistic was suggested by Eubank and Thomas [3] todetect heteroscedasticity of the error term in nonparametricmodels Furthermore Zhang and Mei [7] obtained a test forthe constant variance of the model errors based on residualanalysis

Most of the existing procedures including those men-tioned above belong to the class of parametrically hypothesis

test methods That is the methods work quite well whenthe model errors coincide with the preassumed distributionwhile the performance significantly decreases when thedistribution cannot be guaranteed Therefore it is necessaryto develop a test which is robust to the error distributions Tothe best of our knowledge however there has been little workdone on this issue

In this paper we propose a completely nonparametricallyhypothesis test method for detecting heteroscedasticity of theerror term in nonparametric regression In this method thetest statistic is constructed on the basis of an appropriatetransformation of the residuals after fitting the regressionmodel with the local linear estimation In order to evaluatethe performance of the proposed method we conduct asimulation comparison with Zhang and Meirsquos procedure [7]and a real-world data set is analyzed to show the applicationof the method

The remainder of this paper is organized as followsIn Section 2 we briefly describe the local linear estimationmethod By using the residuals after fitting the regressionmodel with the local linear estimation and applying theidea of trend analysis in nonparametric statistics a testingprocedure is described in Section 3 In Section 4 we conductsome simulations to assess the performance of the test A real-world data set is analyzed in Section 5 to demonstrate theapplication of the proposed methodThe paper is then endedwith some final remarks

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 435925 5 pageshttpdxdoiorg1011552014435925

2 Journal of Applied Mathematics

2 A Brief Description of the LocalLinear Estimation

Consider the univariate nonparametric regression model

119884119894= 119898 (119883

119894) + 120590 (119883

119894) 119890119894 119894 = 1 2 119899 (1)

where 119884 and 119883 indicate the response and explanatoryvariable respectively and (119884

119894 119883119894) (119894 = 1 2 119899) is a

random sample from model (1) 119898(sdot) and 120590(sdot) are unknownregression and variance functions 119890

1 1198902 119890

119899are generally

assumed to be independently and identically distributedrandom variables with zero mean and unit variance Also 119883

and 119890 are independentDue to its several attractive mathematical properties (see

[9ndash11] for details) the local linear estimation procedure isused to calibrate the model in (1) Specifically suppose thatthe second order derivative of the regression function 119898(119909)

in model (1) is continuous in the domain of the variable 119883say D and 119909

0is a given point in D According to Taylorrsquos

expansion we have in the neighborhood of 1199090that

119898(119909) asymp 119898 (1199090) + 1198981015840(1199090) (119909 minus 119909

0) (2)

where1198981015840(1199090) denotes the first order derivative of119898(119909) at 119909

0

By replacing119898(119909) in model (1) with its linear approximationin (2) and combining the least-squares procedure the locallinear estimate of the regression function 119898(119909) at 119909

0can

be obtained by solving the following weighted least-squaresproblem

minimize119899

sum

119894=1

[119884119894minus 119898(119909

0) minus 1198981015840(1199090)(119883119894minus 1199090)]2

119870ℎ(119883119894minus 1199090)

(3)

with respect to 119898(1199090) and 119898

1015840(1199090) where 119870

ℎ(sdot) = 119870(sdotℎ)ℎ

and 119870(sdot) is a given kernel function that is generally takento be a symmetric probability density function and ℎ is thebandwidth which can be determined by some data-drivenmethods such as the cross-validation generalized cross-validation methods and corrected Akaike information crite-rion (see [12ndash14] for more details) Specifically in the cross-validation procedure the optimal value of the bandwidth ℎ ischosen to minimize the following expression

CV (ℎ) =

119899

sum

119894=1

[119884119894minus (119894)(ℎ)]2

(4)

where (119894)(ℎ) stands for the 119894th predicted value of the response

119884 under the bandwidth ℎ with the 119894th observation omittedfrom the calibration process

For convenience we introduce the matrix notations Let

X (0) = (

1 1198831minus 1199090

1 1198832minus 1199090

1 119883119899minus 1199090

) Y = (

1198841

1198842

119884119899

)

120576 = (

1205761

1205762

120576119899

) = (

120590(1198831) 1198901

120590 (1198832) 1198902

120590 (119883119899) 119890119899

)

W (0) = Diag (119870ℎ(1198831minus 1199090)

119870ℎ(1198832minus 1199090) 119870

ℎ(119883119899minus 1199090))

(5)

By solving the weighted least-squares problem in (3) we canobtain the local linear estimate of119898(119909) at 119909 = 119909

0as

(1199090) = eT1[XT

(0)W(0)X(0)]minus1

XT(0)W (0)Y (6)

where e1indicates a two-dimensional vector with its first

element being 1 and the other being 0Taking 119909

0in (6) to be 119883

1 1198832 and 119883

119899 respectively we

can get the fitted value of Y = (1198841 1198842 119884

119899)T denoted by

Y = (1 2

119899)T as

Y = (

(1198831)

(1198832)

(119883119899)

)

= (

eT1[XT

(1)W(1)X(1)]minus1

XT(1)W (1)Y

eT1[XT

(2)W(2)X(2)]minus1

XT(2)W (2)Y

eT1[XT

(119899)W(119899)X(119899)]minus1

XT(119899)W (119899)Y

) = LY

(7)

where

L = (

eT1[XT

(1)W(1)X(1)]minus1

XT(1)W (1)

eT1[XT

(2)W(2)X(2)]minus1

XT(2)W (2)

eT1[XT

(119899)W(119899)X(119899)]minus1

XT(119899)W (119899)

) (8)

is called ldquohatrdquo matrix or smoothing matrixFurther the residual vector can be computed from

= (1205761 1205762 120576

119899)T= Y minus Y = (I minus L)Y (9)

which will be used in the next section

3 A Procedure for DetectingHeteroscedasticity inNonparametric Regression

As mentioned in introduction in real-world data analysiswe rarely know in advance whether the error term is

Journal of Applied Mathematics 3

homoscedastic which deals with the problem of testing forheteroscedasticity That is the hypothesis to be tested is

H0 1205902(119883119894) = 1205902larrrarr H

1 1205902(119883119894) = 1205902 (10)

where 1205902gt 0 is a certain constant

Let = (1205761 1205762 120576

119899)T

= Y minus Y = (I minus L)Y be theresidual vector which is described in (9) In order to constructa test statistic suitable for quantifying the heteroscedasticityof the error term in nonparametric regression we use thetransformed residuals

119903119894=

120576119894

radic2

0ℎ119894119894

119894 = 1 2 119899 (11)

where

2

0=YT

(I minus L)T (I minus L)Ytr [(I minus L)T (I minus L)]

(12)

with ldquotrrdquo standing for the trace of a matrix and ℎ119894119894is the 119894th

diagonal element of the matrixH = (I minus L)T(I minus L)If the null hypothesis H

0in (10) is true which means

that the variance of the error term in model (1) is constantthe values of 119903

2

119894(119894 = 1 2 119899) should not have any

trend whereas there will be some variations in 1199032

1 1199032

2 119903

2

119899

if heteroscedasticity is present Therefore we can test het-eroscedasticity of the error term by analyzing the trend of1199032

119894(119894 = 1 2 119899) Along this line of thinking the hypothesis

in (10) amounts to the hypothesis

H0 1199032

1 1199032

2 119903

2

119899have no trend

larrrarr H1 1199032

1 1199032

2 119903

2

119899have certain trend

(13)

According to the literatures Diblasi and Bowman [15]and Wei et al [16] the random variables 119903

1 1199032 119903

119899are

approximately independent and identically distributed Let

119888 =

119899

2 119899 is even

119899 + 1

2 119899 is odd

1198991015840=

119888 119899 is even119888 minus 1 119899 is odd

119863119894= 1199032

119894minus 1199032

119894+119888 119894 = 1 2 119899

1015840

(14)

Then 1198631 1198632 119863

119899are approximately independent under

H0and 119875H0(119863119894 gt 0) = 119875H0(119863119894 lt 0) = 12 Therefore the

test statistic is constructed as follows

119879 =

119899

sum

119894=1

119868 (119863119894gt 0) (15)

where 119868(sdot) is the indicative functionIf the null hypothesis H

0in (10) (or (13)) is true which

means the model error term is homoscedastic we have

119879 sim 119861(11989910158401

2) (16)

where 119861(1198991015840 12) denotes the binomial distribution with the

parameter being 12 and the sample size being 1198991015840 By noting

the fact that the test statistic 119879 is symmetric or approximatelysymmetric with respect to 119899

10158402 the value of the test statistic

|119879 minus 11989910158402| tends to be large if the error heteroscedasticity is

present Therefore the 119901-value of testing H0versus H

1based

on the statistic 119879 is

119901 = 119875H0 (

100381610038161003816100381610038161003816100381610038161003816

119879 minus1198991015840

2

100381610038161003816100381610038161003816100381610038161003816

ge

100381610038161003816100381610038161003816100381610038161003816

119905 minus1198991015840

2

100381610038161003816100381610038161003816100381610038161003816

)

= 119875H0 (119879 le1198991015840

2minus

100381610038161003816100381610038161003816100381610038161003816

119905 minus1198991015840

2

100381610038161003816100381610038161003816100381610038161003816

) + 119875H0 (119879 ge1198991015840

2+

100381610038161003816100381610038161003816100381610038161003816

119905 minus1198991015840

2

100381610038161003816100381610038161003816100381610038161003816

)

= 2119875H0 (119879 le1198991015840

2minus

100381610038161003816100381610038161003816100381610038161003816

119905 minus1198991015840

2

100381610038161003816100381610038161003816100381610038161003816

)

= 2 times1

21198991015840

(11989910158402)minus|119905minus119899

10158402|

sum

119896=0

119862119896

1198991015840

asymp 2[1 minus Φ(

10038161003816100381610038161003816119879 minus 11989910158402

10038161003816100381610038161003816

radic11989910158402

)]

(17)

where 119905 is the observed value of 119879 computed by (15) For agiven significance level 120572 reject H

0if 119901 lt 120572 otherwise do

not reject H0

4 Simulation Studies

As mentioned in the introduction Zhang and Mei [7] alsoproposed a test method for detecting heteroscedasticity innonparametric models The particular method that theyused is the 119905-test applied to the squared residuals 120576

2

119894(119894 =

1 2 119899) which are shown in (9) A comparisonwithZhangand Meirsquos method is conducted in this section to assess thevalidity of the proposed test method

The following three types of regression and variancefunctions are considered

(1) 119898(119909) = 1 + 119909 120590(119909) = 120590(1 + 119886 sin(8119909))2(2) 119898(119909) = 1 + sin119909 120590(119909) = 120590(4 + 4 119886 cos(4119909))(3) 119898(119909) = 1 + sin119909 120590(119909) = 120590 exp(119886119909)

where 120590 = 05 and 119886 is a constantUsing the above regression and variance functions we can

formulate three models to generate the experimental dataFor convenience the models that correspond to those threesettings of regression and variance functions are denoted byModel 1 Model 2 and Model 3 respectively

In each model the observations 1198831 1198832 119883

119899of the

explanatory variable119883 are equidistantly taken on the interval[0 1] that is 119883

119894= 119894119899 119894 = 1 2 119899 The constant 119886 in

the variance functions is considered to be 0 05 and 10respectively Note that 119886 = 0 refers to the model with theerror term being homoscedastic and the variance functiondeviates from homoscedasticity more and more significantlywith the value of 119886 increasing The sample sizes are taken tobe 119899 = 100 and 119899 = 200 respectively


Table 1 Rejection frequencies of 500 replications of the testing procedure

Model 119886119873(0 1) 119880(minusradic3radic3) 120594

2(4)

Proposed method Zhang andMei [7] Proposed method Zhang and

Mei [7] Proposed method Zhang andMei [7]

119899 = 100

Model 10 0022 0038 0032 0022 0058 0024

050 0346 0014 0402 0002 0348 0026100 0540 0020 0570 0006 0506 0012

Model 20 0020 0038 0038 0024 0048 0030

050 0294 0020 0490 0048 0352 0064100 0996 0032 1000 0056 0996 0080

Model 30 0024 0038 0038 0026 0044 0032

050 0162 0510 0258 0796 0202 0248100 0544 0954 0736 1000 0574 0714

119899 = 200

Model 10 0034 0034 0034 0018 0062 0032

050 0596 0012 0678 0002 0620 0014100 0870 0014 0884 0004 0952 0016

Model 20 0032 0040 0038 0018 0058 0030

050 0436 0022 0690 0052 0514 0068100 1000 0034 1000 0058 1000 0072

Model 30 0038 0042 0038 0018 0058 0030

050 0236 0692 0358 0946 0318 0362100 0714 0996 0912 1000 0750 0862

Furthermore in order to evaluate the robustness of thetest methods (the proposed and Zhang and Meirsquos methods)on the error distributions the random numbers 119890

1 1198902 119890

119899

are independently drawn from 119873(0 1) 119880(minusradic3radic3) andthe standardized Chi-square distribution with 4 degrees offreedom respectively

Given each of Models 1 2 and 3 for each combinationof the values of the constant 119886 the error distributions andthe sample sizes we ran 119873 = 500 attempts of replicationof the testing procedure either for our proposed method orZhang andMeirsquos method in which the Gauss kernel function119870(119909) = exp(minus11990922)radic2120587 is adopted and the bandwidthℎ is selected by the cross-validation procedure Throughout119873 = 500 attempts of replication we record the frequencyof rejecting the null hypothesis under the significance level120572 = 005 and the related results are reported in Table 1

We see fromTable 1 that under the normality distributionof the error term the rejection frequency of both methodsunderH

0(ie 119886 = 0) is reasonably close to the corresponding

significance levels for both sample sizes On the other handtwo test methods perform quite differently for differenttypes of variance functions under the alternative hypothesisAlthough the rejection frequency computed by our methodtends to be undersized for monotone variance function (seeModel 3) it is much larger than that obtained by Zhangand Meirsquos method for high frequency variance functions(see Models 1 and 2) which means that our method is ofsatisfactory power in detecting heteroscedasticity especiallywhen the variance function shows many alternations

Under the situations where the distribution of the modelerror term is nonnormal we see from Table 1 that underH0(ie 119886 = 0) the estimated values of the nominal

probability computed from our method are more stable andmore close to the corresponding significance levels than

those obtained by Zhang and Meirsquos method with respect todifferent types of error distributions which indicates that theproposed test approach is more robust to the choices of theerror distributions Furthermore the values of the rejectionfrequency for both test methods under 119886 = 0 show the samepatterns which demonstrate that our test approach is morepowerful in detecting high frequency variance functions

5 An Example on the Application ofthe Proposed Method

A real-world data set is analyzed in this section to demon-strate the application of the proposed method Specificallywith the observed data of the average temperature (AT) ofeach day inXirsquoan China from January 1 1951 toDecember 312000 the mean of the average temperatures collected on thesame days of the 50 years is taken as the values of the averagetemperature (unit degree) It is worth pointing out that thedata on February 29 during the 50 years have been excludedFurthermore the observations of the explanatory variable 119883

in the regression function are taken as the time orders fromJanuary 1 to December 31

Based on the observations (AT119894 119883119894) (119894 = 1 2 365)

which are graphically shown in Figure 1 the following non-parametric regression model is considered

AT119894= 119898 (119883

119894) + 120590 (119883

119894) 119890119894 119894 = 1 2 365 (18)

where 119890119894is assumed to satisfy E(119890

119894| 119883119894) = 0 and Var(119890

119894|

119883119894) = 1 Here we test whether or not the model error term is

homoscedastic By using the proposed test method with theGauss kernel function the optimal value of the bandwidth ℎ

selected by the cross-validation procedure is ℎ = 2 and theresulting 119901-value is 4713 times 10

minus27 Because of the extremely


0 50 100 150 200 250 300 350 400minus5

0

5

10

15

20

25

30

AT (d

eg)

X (day)

Figure 1 The original data of the response AT in model (18)

small 119901-value of the test statistic we may conclude that theheteroscedasticity of the model error is significant over thetime that ranges from January 1 to December 31

6 Final Remarks

In this paper a test which is free of the types of the errordistributions is developed for detecting heteroscedasticity innonparametric regression models Specifically the statisticis constructed on a basis of appropriate transformation ofthe residuals after fitting the regression model with the locallinear estimation as well as the idea of trend analysis in non-parametric statistics In order to assess the performance of theproposed method we conduct a simulation comparison withother procedures and the results are satisfactory especially forhigh frequency variance functions

Compared to Zhang and Meirsquos method the power ofthe proposed test when heteroscedasticity is present tends tobe underestimated for monotone variance function This isreasonable because the former is mainly formulated based onthe monotone trend of the squared residuals while the latteris a sign-based testing method

Anyhow due to its conceptual simplicity and easy imple-mentation our method is useful in testing heteroscedasticityin nonparametric regression especially for the variancefunctions with many alternations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding publication of this paper

Acknowledgments

This research is supported by the National Natural Sci-ence Foundations of China (no 11326181 and no 11201123)International Cooperative Project in Henan Province (no134300510034) and the start fund of doctorial scientific

research (no 09001624) The authors are especially gratefulto the reviewer and editor for their valuable comments andsuggestions which led to significant improvements in thepaper

References

[1] H Dette ldquoA consistent test for heteroscedasticity in nonpara-metric regression based on the kernel methodrdquo Journal ofStatistical Planning and Inference vol 103 no 1-2 pp 311ndash3292002

[2] H Dette and A Munk ldquoTesting heteroscedasticity in nonpara-metric regressionrdquo Journal of the Royal Statistical Society B vol60 no 4 pp 693ndash708 1998

[3] R L Eubank and W Thomas ldquoDetecting heteroscedasticityin nonparametric regressionrdquo Journal of the Royal StatisticalSociety B vol 55 no 1 pp 145ndash155 1993

[4] H Liero ldquoTesting homoscedasticity in nonparametric regres-sionrdquo Journal of Nonparametric Statistics vol 15 no 1 pp 31ndash512003

[5] J H You and G M Chen ldquoTesting heteroscedasticity inpartially linear regression modelsrdquo Statistics and ProbabilityLetters vol 73 no 1 pp 61ndash70 2005

[6] J H You G M Chen and Y Zhou ldquoStatistical inference ofpartially linear regression models with heteroscedastic errorsrdquoJournal of Multivariate Analysis vol 98 no 8 pp 1539ndash15572007

[7] L Zhang andC-LMei ldquoTesting heteroscedasticity in nonpara-metric regression models based on residual analysisrdquo AppliedMathematics vol 23 no 3 pp 265ndash272 2008

[8] S L Shen C-L Mei and Y J Zhang ldquoSpatially varying coef-ficient models testingfor heteroscedasticity and reweightingestimation of the coefficientsrdquo Environmentand Planning A vol43 no 7 pp 1723ndash1745 2011

[9] J Q Fan ldquoDesign-adaptive nonparametric regressionrdquo Journalof the American Statistical Association vol 87 no 420 pp 998ndash1004 1992

[10] J Q Fan ldquoLocal linear regression smoothers and their minimaxefficienciesrdquo The Annals of Statistics vol 21 no 1 pp 196ndash2161993

[11] D Ruppert andM PWand ldquoMultivariate locally weighted leastsquares regressionrdquo The Annals of Statistics vol 22 no 3 pp1346ndash1370 1994

[12] C M Hurvich J S Simonoff and C L Tsai ldquoSmoothingparameter selection in nonparametric regression using animproved Akaike information criterionrdquo Journal of the RoyalStatistical Society B vol 60 no 2 pp 271ndash293 1998

[13] J D Hart Nonparametric Smoothing and Lack-of-Fit TestsSpringer Series in Statistics Springer New York NY USA 1997

[14] T J Hastie and R J Tibshirani Generalized Additive ModelsChapman and Hall Press London UK 1990

[15] A Diblasi and A Bowman ldquoTesting for constant variance in alinear modelrdquo Statistics and Probability Letters vol 33 no 1 pp95ndash103 1997

[16] B CWei X G Lin and F C Xie Statistical Diagnostics HigherEducation Press Beijing China 2009

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Stochastic AnalysisInternational Journal of


2 A Brief Description of the LocalLinear Estimation

Consider the univariate nonparametric regression model

119884119894= 119898 (119883

119894) + 120590 (119883

119894) 119890119894 119894 = 1 2 119899 (1)

where 119884 and 119883 indicate the response and explanatoryvariable respectively and (119884

119894 119883119894) (119894 = 1 2 119899) is a

random sample from model (1) 119898(sdot) and 120590(sdot) are unknownregression and variance functions 119890

1 1198902 119890

119899are generally

assumed to be independently and identically distributedrandom variables with zero mean and unit variance Also 119883

and 119890 are independentDue to its several attractive mathematical properties (see

[9ndash11] for details) the local linear estimation procedure isused to calibrate the model in (1) Specifically suppose thatthe second order derivative of the regression function 119898(119909)

in model (1) is continuous in the domain of the variable 119883say D and 119909

0is a given point in D According to Taylorrsquos

expansion we have in the neighborhood of 1199090that

119898(119909) asymp 119898 (1199090) + 1198981015840(1199090) (119909 minus 119909

0) (2)

where1198981015840(1199090) denotes the first order derivative of119898(119909) at 119909

0

By replacing119898(119909) in model (1) with its linear approximationin (2) and combining the least-squares procedure the locallinear estimate of the regression function 119898(119909) at 119909

0can

be obtained by solving the following weighted least-squaresproblem

minimize119899

sum

119894=1

[119884119894minus 119898(119909

0) minus 1198981015840(1199090)(119883119894minus 1199090)]2

119870ℎ(119883119894minus 1199090)

(3)

with respect to 119898(1199090) and 119898

1015840(1199090) where 119870

ℎ(sdot) = 119870(sdotℎ)ℎ

and 119870(sdot) is a given kernel function that is generally takento be a symmetric probability density function and ℎ is thebandwidth which can be determined by some data-drivenmethods such as the cross-validation generalized cross-validation methods and corrected Akaike information crite-rion (see [12ndash14] for more details) Specifically in the cross-validation procedure the optimal value of the bandwidth ℎ ischosen to minimize the following expression

CV (ℎ) =

119899

sum

119894=1

[119884119894minus (119894)(ℎ)]2

(4)

where (119894)(ℎ) stands for the 119894th predicted value of the response

119884 under the bandwidth ℎ with the 119894th observation omittedfrom the calibration process

For convenience we introduce the matrix notations Let

X (0) = (

1 1198831minus 1199090

1 1198832minus 1199090

1 119883119899minus 1199090

) Y = (

1198841

1198842

119884119899

)

120576 = (

1205761

1205762

120576119899

) = (

120590(1198831) 1198901

120590 (1198832) 1198902

120590 (119883119899) 119890119899

)

W (0) = Diag (119870ℎ(1198831minus 1199090)

119870ℎ(1198832minus 1199090) 119870

ℎ(119883119899minus 1199090))

(5)

By solving the weighted least-squares problem in (3) we canobtain the local linear estimate of119898(119909) at 119909 = 119909

0as

(1199090) = eT1[XT

(0)W(0)X(0)]minus1

XT(0)W (0)Y (6)

where e1indicates a two-dimensional vector with its first

element being 1 and the other being 0Taking 119909

0in (6) to be 119883

1 1198832 and 119883

119899 respectively we

can get the fitted value of Y = (1198841 1198842 119884

119899)T denoted by

Y = (1 2

119899)T as

Y = (

(1198831)

(1198832)

(119883119899)

)

= (

eT1[XT

(1)W(1)X(1)]minus1

XT(1)W (1)Y

eT1[XT

(2)W(2)X(2)]minus1

XT(2)W (2)Y

eT1[XT

(119899)W(119899)X(119899)]minus1

XT(119899)W (119899)Y

) = LY

(7)

where

L = (

eT1[XT

(1)W(1)X(1)]minus1

XT(1)W (1)

eT1[XT

(2)W(2)X(2)]minus1

XT(2)W (2)

eT1[XT

(119899)W(119899)X(119899)]minus1

XT(119899)W (119899)

) (8)

is called ldquohatrdquo matrix or smoothing matrixFurther the residual vector can be computed from

= (1205761 1205762 120576

119899)T= Y minus Y = (I minus L)Y (9)

which will be used in the next section

3 A Procedure for DetectingHeteroscedasticity inNonparametric Regression

As mentioned in introduction in real-world data analysiswe rarely know in advance whether the error term is



H0 1205902(119883119894) = 1205902larrrarr H

1 1205902(119883119894) = 1205902 (10)


Let = (1205761 1205762 120576

119899)T


119903119894=

120576119894

radic2

0ℎ119894119894

119894 = 1 2 119899 (11)

where

2

0=YT


(12)





2



1 1199032

2 119903

2

119899




H0 1199032

1 1199032

2 119903

2

119899have no trend

larrrarr H1 1199032

1 1199032

2 119903

2


(13)


1 1199032 119903

119899are


119888 =

119899

2 119899 is even

119899 + 1

2 119899 is odd

1198991015840=


119863119894= 1199032

119894minus 1199032

119894+119888 119894 = 1 2 119899

1015840

(14)

Then 1198631 1198632 119863




119879 =

119899

sum

119894=1

119868 (119863119894gt 0) (15)




119879 sim 119861(11989910158401

2) (16)







1based


119901 = 119875H0 (

100381610038161003816100381610038161003816100381610038161003816

119879 minus1198991015840

2

100381610038161003816100381610038161003816100381610038161003816

ge

100381610038161003816100381610038161003816100381610038161003816

119905 minus1198991015840

2

100381610038161003816100381610038161003816100381610038161003816

)

= 119875H0 (119879 le1198991015840

2minus

100381610038161003816100381610038161003816100381610038161003816

119905 minus1198991015840

2

100381610038161003816100381610038161003816100381610038161003816

) + 119875H0 (119879 ge1198991015840

2+

100381610038161003816100381610038161003816100381610038161003816

119905 minus1198991015840

2

100381610038161003816100381610038161003816100381610038161003816

)

= 2119875H0 (119879 le1198991015840

2minus

100381610038161003816100381610038161003816100381610038161003816

119905 minus1198991015840

2

100381610038161003816100381610038161003816100381610038161003816

)

= 2 times1

21198991015840

(11989910158402)minus|119905minus119899

10158402|

sum

119896=0

119862119896

1198991015840

asymp 2[1 minus Φ(

10038161003816100381610038161003816119879 minus 11989910158402

10038161003816100381610038161003816

radic11989910158402

)]

(17)



not reject H0



2

119894(119894 =







119899of the


119894= 119894119899 119894 = 1 2 119899 The constant 119886 in





2(4)



119899 = 100

Model 10 0022 0038 0032 0022 0058 0024

050 0346 0014 0402 0002 0348 0026100 0540 0020 0570 0006 0506 0012

Model 20 0020 0038 0038 0024 0048 0030

050 0294 0020 0490 0048 0352 0064100 0996 0032 1000 0056 0996 0080

Model 30 0024 0038 0038 0026 0044 0032

050 0162 0510 0258 0796 0202 0248100 0544 0954 0736 1000 0574 0714

119899 = 200

Model 10 0034 0034 0034 0018 0062 0032

050 0596 0012 0678 0002 0620 0014100 0870 0014 0884 0004 0952 0016

Model 20 0032 0040 0038 0018 0058 0030

050 0436 0022 0690 0052 0514 0068100 1000 0034 1000 0058 1000 0072

Model 30 0038 0042 0038 0018 0058 0030

050 0236 0692 0358 0946 0318 0362100 0714 0996 0912 1000 0750 0862


1 1198902 119890

119899














AT119894= 119898 (119883

119894) + 120590 (119883

119894) 119890119894 119894 = 1 2 365 (18)


119894| 119883119894) = 0 and Var(119890

119894|






0 50 100 150 200 250 300 350 400minus5

0

5

10

15

20

25

30

AT (d

eg)

X (day)



6 Final Remarks






Acknowledgments



References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











H0 1205902(119883119894) = 1205902larrrarr H

1 1205902(119883119894) = 1205902 (10)


Let = (1205761 1205762 120576

119899)T


119903119894=

120576119894

radic2

0ℎ119894119894

119894 = 1 2 119899 (11)

where

2

0=YT


(12)





2



1 1199032

2 119903

2

119899




H0 1199032

1 1199032

2 119903

2

119899have no trend

larrrarr H1 1199032

1 1199032

2 119903

2


(13)


1 1199032 119903

119899are


119888 =

119899

2 119899 is even

119899 + 1

2 119899 is odd

1198991015840=


119863119894= 1199032

119894minus 1199032

119894+119888 119894 = 1 2 119899

1015840

(14)

Then 1198631 1198632 119863




119879 =

119899

sum

119894=1

119868 (119863119894gt 0) (15)




119879 sim 119861(11989910158401

2) (16)







1based


119901 = 119875H0 (

100381610038161003816100381610038161003816100381610038161003816

119879 minus1198991015840

2

100381610038161003816100381610038161003816100381610038161003816

ge

100381610038161003816100381610038161003816100381610038161003816

119905 minus1198991015840

2

100381610038161003816100381610038161003816100381610038161003816

)

= 119875H0 (119879 le1198991015840

2minus

100381610038161003816100381610038161003816100381610038161003816

119905 minus1198991015840

2

100381610038161003816100381610038161003816100381610038161003816

) + 119875H0 (119879 ge1198991015840

2+

100381610038161003816100381610038161003816100381610038161003816

119905 minus1198991015840

2

100381610038161003816100381610038161003816100381610038161003816

)

= 2119875H0 (119879 le1198991015840

2minus

100381610038161003816100381610038161003816100381610038161003816

119905 minus1198991015840

2

100381610038161003816100381610038161003816100381610038161003816

)

= 2 times1

21198991015840

(11989910158402)minus|119905minus119899

10158402|

sum

119896=0

119862119896

1198991015840

asymp 2[1 minus Φ(

10038161003816100381610038161003816119879 minus 11989910158402

10038161003816100381610038161003816

radic11989910158402

)]

(17)



not reject H0



2

119894(119894 =







119899of the


119894= 119894119899 119894 = 1 2 119899 The constant 119886 in





2(4)



119899 = 100

Model 10 0022 0038 0032 0022 0058 0024

050 0346 0014 0402 0002 0348 0026100 0540 0020 0570 0006 0506 0012

Model 20 0020 0038 0038 0024 0048 0030

050 0294 0020 0490 0048 0352 0064100 0996 0032 1000 0056 0996 0080

Model 30 0024 0038 0038 0026 0044 0032

050 0162 0510 0258 0796 0202 0248100 0544 0954 0736 1000 0574 0714

119899 = 200

Model 10 0034 0034 0034 0018 0062 0032

050 0596 0012 0678 0002 0620 0014100 0870 0014 0884 0004 0952 0016

Model 20 0032 0040 0038 0018 0058 0030

050 0436 0022 0690 0052 0514 0068100 1000 0034 1000 0058 1000 0072

Model 30 0038 0042 0038 0018 0058 0030

050 0236 0692 0358 0946 0318 0362100 0714 0996 0912 1000 0750 0862


1 1198902 119890

119899














AT119894= 119898 (119883

119894) + 120590 (119883

119894) 119890119894 119894 = 1 2 365 (18)


119894| 119883119894) = 0 and Var(119890

119894|






0 50 100 150 200 250 300 350 400minus5

0

5

10

15

20

25

30

AT (d

eg)

X (day)



6 Final Remarks






Acknowledgments



References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












2(4)



119899 = 100

Model 10 0022 0038 0032 0022 0058 0024

050 0346 0014 0402 0002 0348 0026100 0540 0020 0570 0006 0506 0012

Model 20 0020 0038 0038 0024 0048 0030

050 0294 0020 0490 0048 0352 0064100 0996 0032 1000 0056 0996 0080

Model 30 0024 0038 0038 0026 0044 0032

050 0162 0510 0258 0796 0202 0248100 0544 0954 0736 1000 0574 0714

119899 = 200

Model 10 0034 0034 0034 0018 0062 0032

050 0596 0012 0678 0002 0620 0014100 0870 0014 0884 0004 0952 0016

Model 20 0032 0040 0038 0018 0058 0030

050 0436 0022 0690 0052 0514 0068100 1000 0034 1000 0058 1000 0072

Model 30 0038 0042 0038 0018 0058 0030

050 0236 0692 0358 0946 0318 0362100 0714 0996 0912 1000 0750 0862


1 1198902 119890

119899














AT119894= 119898 (119883

119894) + 120590 (119883

119894) 119890119894 119894 = 1 2 365 (18)


119894| 119883119894) = 0 and Var(119890

119894|






0 50 100 150 200 250 300 350 400minus5

0

5

10

15

20

25

30

AT (d

eg)

X (day)



6 Final Remarks






Acknowledgments



References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 250 300 350 400minus5

0

5

10

15

20

25

30

AT (d

eg)

X (day)



6 Final Remarks






Acknowledgments



References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Testing Heteroscedasticity in...

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