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Bivariate Sign Test for One-SampleBivariate Location Model Using RankedSet SampleHani M. Samawi a , Mohammad F. Al-Saleh a & Obaid Al-Saidy ba Department of Statistics , Yarmouk University , Irbid, Jordanb Department of Mathematics and Statistics , Sultan QaboosUniversity , Al-khod, Sultanate of OmanPublished online: 15 Feb 2007.

To cite this article: Hani M. Samawi , Mohammad F. Al-Saleh & Obaid Al-Saidy (2006) Bivariate SignTest for One-Sample Bivariate Location Model Using Ranked Set Sample, Communications in Statistics- Theory and Methods, 35:6, 1071-1083, DOI: 10.1080/03610920600579929

To link to this article: http://dx.doi.org/10.1080/03610920600579929

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Communications in Statistics—Theory and Methods, 35: 1071–1083, 2006Copyright © Taylor & Francis Group, LLCISSN: 0361-0926 print/1532-415X onlineDOI: 10.1080/03610920600579929

Nonparametric Inference

Bivariate Sign Test for One-Sample BivariateLocationModel Using Ranked Set Sample

HANI M. SAMAWI1, MOHAMMAD F. AL-SALEH1,AND OBAID AL-SAIDY2

1Department of Statistics, Yarmouk University, Irbid, Jordan2Department of Mathematics and Statistics, Sultan Qaboos University,Al-khod, Sultanate of Oman

In this article, we introduce a bivariate sign test for the one-sample bivariatelocation model using a bivariate ranked set sample (BVRSS). We show that theproposed test is asymptotically more efficient than its counterpart sign test based ona bivariate simple random sample (BVSRS). The asymptotic null distribution and thenon centrality parameter are derived. The asymptotic distribution of the vector ofsample median as an estimator of the locations of the bivariate model is introduced.Theoretical and numerical comparisons of the asymptotic efficiency of the BVRSSsign test with respect to the BVSRS sign test are also given.

Keywords Bivariate ranked set sample; One-sample bivariate location model;Power of the test; Ranked set sample; Sign test.

Mathematics Subject Classification 62G10.

1. Introduction

In many agricultural and environmental studies, and recently even in humanpopulation studies, it is possible to rank the experimental or sampling units withrespect to the variable of interest, without actually measuring these units. Thismethod of sampling was first proposed by McIntyre (1952) for estimating the meanyield in pasture. Halls and Dell (1966) named the procedure ranked set sampling(RSS).

The RSS procedure can be described as follows: identify a group of r2 samplingunits from the target population. Then, partition the identified units randomly intor disjoint subsets each having a pre-assigned size r. In most practical situations,the size r is 2, 3, or 4. Each subset is then ranked by a suitable ranking method(e.g., prior information, visual inspection or by the subject-matter experimenter

Received March 9, 2005; Accepted October 28, 2005Address correspondence to Hani M. Samawi, Department of Statistics, Yarmouk

University, Irbid 211-63, Jordan; E-mail: hsamawi@yu.edu.jo

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1072 Samawi et al.

himself, � � � etc.) The ith ordered statistic from the ith subset, Xi�i�, i = 1� � � � � r, isquantified (actual measurement) and X1�1�� X2�2�� � � � � Xr�r� constitute the RSS. Thisrepresents one cycle. The whole procedure can be repeated m times as needed to geta RSS of size n = mr.

Stokes and Sager (1988) gave estimators of distribution functions usingempirical distribution functions based on RSS �F ∗�. Quantiles estimation using RSSwere discussed by Chen (2000) and Samawi (2001). An optimal ranked set samplescheme (ORSS) for inference on population quantiles was suggested by Chen (2001).Nonparametric procedures based on RSS are considered by Bohn and Wolfe (1992,1994), Kvam and Samaniego (1994), and Hettmansperger (1995). For the theoreticalaspects of RSS see Takahasi and Wakimoto (1968) or Dell and Clutter (1977).Variations of RSS are given by Kaur et al. (1995) and Patil et al. (1999).

Koti and Babu (1996) studied the exact distribution of the RSS sign test. Theyshowed that this test provides a more powerful test than the SRS sign test. Barabesi(1998) provides a simpler and faster method for computing the exact distribution ofthe RSS sign test. The optimality of RSS sign test has been established via Pitmanasymptotic efficacy by several researchers in the literature. It is shown that themedian ranked set sample (MRSS) is the best sampling schemes in the ranked setsampling environment for the sign test procedure; for example see Ozturk (1999)and Ozturk and Wolfe (2000). To our knowledge, however, the optimality of theMRSS for the sign test has only been shown asymptotically. Samawi and Abu-Dayyeh (2003) investigated the exact power and distribution function for finitesample sizes of the MRSS sign test.

Multiple characteristics estimation using a bivariate ranked set sample (BVRSS)was introduced by Al-Saleh and Zheng (2002). Based on their procedure, a BVRSScan be obtained as follows:

Suppose �X� Y � is a bivariate random vector with the joint probability densityfunction (pdf) f�x� y�. To obtain a BVRSS, we follow the following steps:

1. A random sample of size r4 is identified from the population and randomlyallocated into r2 pools each of size r2 so that each pool is a square matrix with rrows and r columns.

2. In the first pool, identify the minimum value by judgment w.r.t. the firstcharacteristic X, for each of the r rows.

3. For the r minima obtained in Step 2, the actual quantification is done onthe pair that corresponds to the minimum value of the second characteristic, Y ,identified by judgment. This pair, resembles the label (1,1), is the first element of theBVRSS sample.

4. Repeat Steps 2 and 3 for the second pool, but in Step 3, the pair thatcorresponds to the second minimum value w.r.t. the second characteristic, Y , ischosen for actual quantification. This pair resembles the label (1, 2).

5. The process continues until the label �r� r� is resembled from the r2th (last)pool.

The procedure described above produces a BVRSS of size r2. The procedurecan be repeated m times to obtain a BVRSS sample of size n = mr2. Although theBVRSS procedure can easily be extended to multivariate RSS (MVRSS), to simplifythe presentation of the proposed method herein, we only consider the bivariate case(BVRSS).

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Bivariate Ranked Set Sampling sign Test 1073

In this article we introduce the bivariate ranked set sample sign test for the one-sample bivariate location model. Numerical comparisons between the performanceof the BVRSS sign test and the performance of the BVSRS sign test via theirasymptotic efficiencies are investigated. The asymptotic null distribution and thenon centrality parameter of the alternative distribution of the proposed test arederived. It is shown that BVRSS substantially improves the efficiency of the sign testfor the one-sample bivariate location model. We will also demonstrate that usingBVRSS for estimating the locations of the bivariate model is more efficient thanusing BVSRS.

1.1. Characteristics of BVRSS and Some Useful Results

Let ��Xzijk� Y

zijk�, i = 1� 2� � � � � r, j = 1� 2� � � � � r, k = 1� 2� � � � � m & z = 1� � � � � r2� be

mr4 i.i.d ordered pairs from a bivariate probability density function, say f�x� y���x� y� ∈ R2. Following the Al-Saleh and Zheng (2002) definition of BVRSS let[(Xz

�i��j�k� Yz�i��j�k

)� i= 1� 2� � � � � r� j= 1� 2� � � � � r� z= �j − 1�r + i and k= 1� 2� � � � � m

]denotes such a sample from f�x� y�. Let fXz

�i��j��Yz�i��j�

�x� y� be the joint pdf of(Xz

�i��j�k� Yz�i��j�k

), k = 1� 2� � � � � m. Then as in Al-Saleh and Zheng (2002),

fXz�i��j��Y

z�i��j�

�x� y� = fY�i��j� �y�fX�j��x�fY �X�y�x�

fY�j� �y�(1.1)

where fX�j�is the density of the jth order statistic for a SRS sample of size r from the

marginal density of fX� fY�j� �y� is the density of the corresponding Y -value given by

fY�j� �y� =∫ �

−�fX�j��x�fY �X�y�x�dx�

and fY�i��j� �y� is the density of the ith order statistic of an i.i.d. sample from fY�j� �y�,i.e.,

fY�i��j� �y� = c(FY�j�

�y�)i−1(

1− FY�j��y�)r−i

fY�j� �y�

with FY�j��y� = ∫ y

−�( ∫ �

−� fX�j��x�fY �X�w�x�dx

)dw and c = r!

�i−1�!�r−i�! .Using the above results and dropping the z notation for simplicity, we rewrite

Eq. (1.1) as

fX�i��j��Y�i��j��x� y� = c1

(FY�j�

�y��i−1�1− FY�j��y��r−i�FX�x��

j−1�1− FX�x��r−jf�x� y� (1.2)

where

c1 =r!

�i− 1�!�r − i�!r!

�j − 1�!�r − j�! �

Again from Al-Saleh and Zheng (2002) we have the following results:

�1�1r2

r∑j=1

r∑i=1

fX�i��j��Y�i��j��x� y� = f�x� y�� (1.3)

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1074 Samawi et al.

�2�1r2

r∑j=1

r∑i=1

fX�i��j��x� = fX�x�� (1.4)

�3�1r2

r∑j=1

r∑i=1

fY�i��j� �y� = fY �y�� (1.5)

Also, see Chen et al. (2004) and Chen and Shen (2003) for the proof in the generalcase including imperfect ranking. However, in this article we assume perfect rankingin both variables X and Y .

1.2. General Setting

Suppose that the bivariate random variable �X� Y � has an absolutely continuouscumulative distribution function (cdf) F�x − �1� y − �2� with absolutely continuousmarginal cdfs FX�x − �1� and FY �y − �2�, where the location parameters �1 and�2 are the marginal medians and Fx� Fy ∈ 0 = F � F absolutely continuous andF�0� = 1

2 � uniquely}, see for example Hettmansperger (1984).We are then interested in testing H0 � �˜

= 0˜versus HA � �

˜�= 0

˜, where �

˜= [

�1�2

]and 0

˜= [

00

]. The hypothesis H0 � �˜

= �˜can be transformed into H0 � �˜

= 0˜by

subtracting the given �˜from the observation vectors.

2. Sign Test for One-Sample Bivariate Location Model

2.1. Sign Test Based on BVSRS

Hettmansperger (1984) showed that, since no further structural assumptions onF�X� Y � was made, the vector of sign statistics is appropriate for the constructionof a test for the hypotheses given above. The form of the sign statistics withzero expectation under the null hypothesis is considered in this article. Let�X1� Y1�� �X2� Y2�� � � � � �Xn� Yn� be a BVSRS from F with n = mr2. Then the BVSRSsign test statistic is given by

S′˜=(Sx =

n∑i=1

sgnXi = 2n∑

i=1

I�Xi > 0�− n a�e��

Sy =n∑

i=1

sgn Yi = 2n∑

i=1

I�Yi > 0�− n a�e�

)(2.1)

where I�·� is the indicator function defined by:

I�t > 0� ={1 if t > 0

0 otherwise�

and sgn�t� = −1 if t < 0, 0 if t = 0, and +1 if t > 0. The asymptotic distributionfunction of S′

˜is given by Hettmansperger (1984) as follows: Under the null

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Bivariate Ranked Set Sampling sign Test 1075

hypothesis H0 � �˜= 0

˜, 1√

nS′˜

D−→N2

(0˜� V), where V =

(1 vxyvxy 1

)and

vxy = P�X ≤ 0� Y ≤ 0�+ P�X > 0� Y > 0�− P�X ≤ 0� Y > 0�− P�X > 0� Y ≤ 0�(2.2)

or

vxy = 4P�X ≤ 0� Y ≤ 0�− 1�

From (2.2) and the law of large numbers, a consistent estimate of vxy is given by

v̂xy =1n

n∑i=1

sgnXi sgnYi� (2.3)

Then, under the null hypothesis, a consistent estimate of V is V̂ =(

1 v̂xyv̂xy 1

)and the

test statistics

SBVSRS = S′˜�nV̂ �−1

S˜

(2.4)

is asymptotically distributed as �2�2�.A practical and simple form of SBVSRS is given by

SBVSRS =�N00 − N11�

2

N00 + N11

+ �N01 − N10�2

N01 + N10

� (2.5)

where N00 = #�Xi ≤ 0� Yi ≤ 0�, N11 = #�Xi > 0� Yi > 0�, N01 = #�Xi ≤ 0� Yi > 0� andN10 = #�Xi > 0� Yi ≤ 0�, i = 1� 2� � � � � n, see Hettmansperger (1984).

2.2. Sign Test Based on BVRSS

Using the BVRSS sample,[(X�i��j�k� Y�i��j�k

)� i = 1� 2� � � � � r� j = 1� 2� � � � � r� and k = 1� 2� � � � � m

]�

drawn from F , the BVRSS sign test statistic can be defined by

S′˜R

= �SxR� SyR�� (2.6)

where,

SxR =r∑

i=1

r∑j=1

m∑k=1

sgnX�i��j�k = 2r∑

i=1

r∑j=1

m∑k=1

I(X�i��j�k > 0

)− n

a.e. and

SyR =r∑

i=1

r∑j=1

m∑k=1

sgnY�i��j�k = 2r∑

i=1

r∑j=1

m∑k=1

I(Y�i��j�k > 0

)− n

a.e.

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1076 Samawi et al.

Theorem 2.1. Under the null hypothesis H0 � �˜= 0

˜, for a fixed value of r and a large

value of m, and hence for a large n

1√nS′˜R

D−→N2

(0˜� VR

)�

where

VR =(vxxR vxyRvxyR vyyR

)� n = mr2�

vxxR = 2[1− 2

r2

r∑i=1

r∑j=1

F 2X�i��j�

�0�]� vyyR = 2

[1− 2

r2

r∑i=1

r∑j=1

F 2Y�i��j�

�0�]�

and

vxyR = 4[P�X > 0� Y > 0�− 1

r2

r∑i=1

r∑j=1

FX�i��j��0�FY�i��j�

�0�]�

Proof. First

E�SxR� =r∑

i=1

r∑j=1

m∑k=1

E�sgnX�i��j�k� = 2r∑

i=1

r∑j=1

m∑k=1

E[I�X�i��j�k > 0�

]− n

= 2r∑

i=1

r∑j=1

m∑k=1

[1− FX�i��j�

�0�]− n = 2mr2 − 2mr2FX�0�−mr2 = 0�

by (1.4) and FX�0� = 12 . Using a similar argument it is easily shown that E�SyR� = 0.

Using the fact that FX�0� = 12 and by (1.4), the variance of SxR is

V�SxR� =r∑

i=1

r∑j=1

m∑k=1

V �sgnX�i��j�k� = 4r∑

i=1

r∑j=1

m∑k=1

V[I�X�i��j�k > 0�

]= 4

[ r∑i=11

r∑j=1

m∑k=1

[1− FX�i��j�

�0�]FX�i��j�

�0�]= 2n

[1− 2

r2

r∑i=1

r∑j=1

F 2X�i��j�

�0�]�

Similarly, it can be shown that V�SyR� = 2n[1− 2

r2

∑ri=1

∑rj=1 F

2Y�i��j�

�0�]and

Cov�SxR� SyR� = 4n[P�X > 0� Y > 0�− 1

r2

r∑i=1

r∑j=1

FX�i��j��0�FY�i��j�

�0�]�

Next, since the BVRSS sign test statistic S′˜R

= �SxR� SyR� components can beredefined by the sums of i.i.d. variates, as

SxR = 2r∑

i=1

r∑j=1

m∑k=1

I�X�i��j�k > 0�− n =m∑

k=1

SxRk and

SyR = 2r∑

i=1

r∑j=1

m∑k=1

I�Y�i��j�k > 0�− n =m∑

k=1

SyRk�

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Bivariate Ranked Set Sampling sign Test 1077

where

SxRk = 2r∑

i=1

r∑j=1

I�X�i��j�k > 0�− r2� with E�SxRk� = 0 and

V�SxRk� = 2r2[1− 2

r2

r∑i=1

r∑j=1

F 2X�i��j�

�0�]< �

and

SyRk = 2r∑

i=1

r∑j=1

I�Y�i��j�k > 0�− r2� with E�SyRk� = 0 and

V�SyRk� = 2r2[1− 2

r2

r∑i=1

r∑j=1

F 2Y�i��j�

�0�]< �

k = 1� 2� � � � � m, for fixed set size, r. Therefore, Theorem A12 in Hettmansperger(1984) implies that the limiting distribution for 1√

nS′˜R

is N2

(0˜� VR

).

By the law of large numbers and the results in Sec. 1.1: As m→�, we have that

�1� F̂X�i��j�= 1

m

m∑k=1

I�X�i��j�k ≤ 0�P−→FX�i��j�

�

�2� F̂Y�i��j�= 1

m

m∑k=1

I�Y�i��j� ≤ 0�P−→FY�i��j�

�3� P̂�X > 0� Y > 0� = 1n

m∑k=1

r∑i=1

r∑j=1

I(X�i��j�k > 0� Y�i��j� > 0

) P−→P�X > 0� Y > 0��

Thus a consistent estimator of VR is

V̂R =(v̂xxR v̂xyR

v̂xyR v̂yyR

)�

where

v̂xxR = 2

[1− 2

r2

r∑i=1

r∑j=1

F̂ 2X�i��j�

�0�

]� v̂yyR = 2

[1− 2

r2

r∑i=1

r∑j=1

F̂ 2Y�i��j�

�0�

]

and

v̂xyR = 4

[1n

m∑k=1

r∑i=1

r∑j=1

I(X�i��j�k > 0� Y�i��j�k > 0

)− 1r2

r∑i=1

r∑j=1

F̂X�i��j��0�F̂Y�i��j��0�

]�

Using these results, the test statistic

SBVRSS = S′˜R�nV̂R�

−1 S′˜R

(2.7)

is asymptotically distributed as �2�2�.

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1078 Samawi et al.

Now define the two-way table of counts as follows: For i� j = 1� 2� � � � � r,

Y�i��j�

X�i��j� ≤0 >0 Totals

≤0 Nij 00 Nij 01 Nij 0∗

>0 Nij 10 Nij 11 Nij 1∗

Totals Nij∗0 Nij∗1 Nij∗∗

Also, define the following:

N00 =r∑

i=1

r∑j=1

Nij 00� N10 =r∑

i=1

r∑j=1

Nij 10� N01 =r∑

i=1

r∑j=1

Nij 01�

N11 =r∑

i=1

r∑j=1

Nij 11 and n = mr2 = N = N00 + N01 + N10 + N11�

Using these notations, S′˜R

= �SxR� SyR� can be written as

S′˜R

= ��N11 − N00�+ �N10 − N01�� �N11 − N00�− �N10 − N01��

and �nV̂R�−1 as

�nV̂R�−1 =

(v1R v2Rv2R v3R

)�

where

v1R = 1− 2∑r

i=1∑r

j=1 N2ij∗0

m2r2

2nD� v2R =

−(

N11r2

r2−2

∑ri=1

∑rj=1 Nij∗0 Nij0∗m2r2

)nD

� v3R = 1− 2∑r

i=1∑r

j=1 N2ij0∗

m2r2

2nD

and

D =[(

1− 2∑r

i=1

∑rj=1 N

2ij0∗

m2r2

)(1− 2

∑ri=1

∑rj=1 N

2ij∗0

m2r2

)

−4(N11

r2− 2

∑ri=1

∑rj=1 Nij∗0Nij 0∗

m2r2

)2]�

Hence,

SBVRSS = S′˜R�nV̂R�

−1S˜R= �N11 − N00�

2�v1R + 2v2R + v3R�

+ 2�N11 − N00��N10 − N01��v1R − v3R�

+ �N10 − N01�2�v1R − 2v2R + v3R�� �

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Bivariate Ranked Set Sampling sign Test 1079

3. Asymptotic Efficiency

Hettmansperger (1984) argued that the asymptotic efficiency of the BVSRS signtest can be obtained in a manner similar to that of the univariate case as follows:Since along a sequence of alternatives �

˜ n= �∼√

nconverging to 0

˜, E�n

( S∼√n

)converges

to ˜= 2��1fX�0�� �2fY �0��

′. Thus, along a sequence of alternatives �˜ n,

S∼√nhas an

asymptotic distribution, N2� ˜� V�, where V is as defined in Sec. 2.1. Therefore, the

limiting distribution of the statistic SBVSRS , along �˜ n, will be a non central chi-square

with two degrees of freedom, and the non centrality parameter is ′˜V−1

˜. Similarly,

to establish the limiting distribution of 1√nS′˜R

along �˜ n, when using BVRSS, note that

E�n

(1√n S˜R

) = E�n

(1√n S˜

). Hence, along the same sequence of alternative �

˜ n, E�n

(1√n S˜R

)converges to

˜= 2��1fX�0�� �2fY �0��

′. Thus, using the notation of Theorem 2.1, the

limiting distribution of the statistic SBVRSS , along �˜ n, will be a non central chi-square

with two degrees of freedom, and a non centrality parameter ′˜V−1R

˜.

Moreover, since the asymptotic local power is an increasing function of thenoncentrality parameter, the efficiency of the two test statistics can be obtained asthe ratio of their noncentrality parameters. Hence,

eff �SBVRSS� SBVSRS� =[ ′

˜V−1R ′

˜ ′˜V−1R ′

˜

]� (3.1)

Moreover, it is well-known that

inf

′˜V−1R ′

˜ ′˜V−1R ′

˜

= ��1� and sup

′˜V−1R ′

˜ ′˜V−1� ′

˜

= ��2�

where ��1� ≤ ��2� denote the ordered eigenvalues of the matrix VV−1R . See, for example

Seber (1984). Now, to show the superiority of SBVRSS over SBVSRS , next we prove thefollowing result.

Theorem 3.1. Under the null hypothesis H0 � �˜= 0

˜, and the assumptions in Sec. (1.2),

eff �SBVRSS� SBVSRS� ≥ 1.

Proof. It is suffices to show that �1� �2 ≥ 1 where �1 and �2 are the eigenvalues ofthe matrix VV−1

R . Using simple linear algebra we have the following results:

�1� det�VV−1R � = det�V�det�V−1

R � = det�V�det�VR�

= �1�2 =�1− v2xy�

vxxRvyyR − v2xyR�

�2� tr�VV−1R � = �1 + �2 =

vxxR + vyyR − 2vxyRvxyvxxRvyyR − v2xyR

�

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1080 Samawi et al.

Noting that vxxR = vyyR and solving (1) and (2) for �1 and �2 simultaneously, we get

�1 =�1− vxy�

�vxxR − vxyR�

= 2− 4P�X ≤ 0� Y ≤ 0�

2− P�X > 0� Y > 0�+ 4(

1r2

∑ri=1

∑rj=1 FX�i��j�

FY�i��j�− 1

r2

∑ri=1

∑rj=1 F

2X�i��j�

)and

�2 =�1+ vxy�

�vxxR + vxyR�

= 4P�X ≤ 0� Y ≤ 0�

2+ 4P�X > 0� Y > 0�− 4(

1r2

∑ri=1

∑rj=1 FX�i��j�

FY�i��j�+ 1

r2

∑ri=1

∑rj=1 F

2X�i��j�

) �Since, FX�0� = FY �0�, it can be shown that P�X ≤ 0� Y ≤ 0� = P�X > 0� Y > 0�. Also,since 1

r2

∑ri=1

∑rj=1 F

2X�i��j�

= 1r2

∑ri=1

∑rj=1 F

2Y�i��j�

, then by using the discrete type ofSchwarz’s inequality, we have

1r2

r∑i=1

r∑j=1

FX�i��j�FY�i��j�

≤(1r2

r∑i=1

r∑j=1

F 2X�i��j�

1r2

r∑i=1

r∑j=1

F 2Y�i��j�

)1/2

= 1r2

r∑i=1

r∑j=1

F 2Y�i��j�

�

Therefore, it is clear that �1 ≥ 1. Similarly, using the results in (1.4) and (1.5), thefact that F�·� is monotone increasing function, Jensen’s inequality, and the discretetype of covariance inequality we have the following:

�i�1r2

r∑i=1

r∑j=1

F 2X�i��j�

≥(1r2

r∑i=1

r∑j=1

FX�i��j�

)2

= F 2X�0� =

14�

�ii�1r2

r∑i=1

r∑j=1

FX�i��j�FY�i��j�

≥(1r2

r∑i=1

r∑j=1

FX�i��j�

1r2

r∑i=1

r∑j=1

FY�i��j�

)= FX�0�FY �0� =

14�

Hence,

2+ 4P�X > 0� Y > 0�− 4(1r2

r∑i=1

r∑j=1

FX�i��j�FY�i��j�

+ 1r2

r∑i=1

r∑j=1

F 2X�i��j�

)≤ 4P�X > 0� Y > 0��

Therefore, �2 ≥ 1.For more extensive analysis of the efficiency properties of the multivariate sign

test, see Bickel (1965). Also, see Bai and Chen (2003).

4. Asymptotic Distribution of the Sample Medians

The following result is from Hettmansperger (1984): Let�X1� Y1�, �X2� Y2�� � � � ��Xn� Yn�, be a BVRS from the distribution F , with n = mr2, and let �̂′

˜= ��̂1� �̂2� be

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Bivariate Ranked Set Sampling sign Test 1081

the vector of sample medians of X and Y , respectively, i.e., �̂1 = med�X1� X2� � � � � Xn�

and �̂2 = med�Y1� Y2� � � � � Yn�. Then√n �̂

˜D−→N2�0˜

�∑�, where

∑ = 1

4f 2X�0�4P�X< 0�Y< 0�−1

4fX�0�fY �0�

4P�X< 0�Y< 0�−14fX�0�fY �0�

14f 2X�0�

�

Using a similar argument as in Hettmansperger (1984) and the results in Sec. 2, itcan be shown that the BVRSS vector of sample medians of X and Y , �̂′

˜ R= �̂1R =

Medijk�X�i��j�k�, �̂2R = Medijk�Y�i��j�k�, i� j = 1� 2� � � � � r, k = 1� 2� � � � � m�, also has a

limiting bivariate normal distribution, that is,√n �̂

˜RD−→N2�0˜

�∑

R�, where

∑R

=

1− 2r2∑r

i=1∑r

j=1 F2X�i��j�

�0�

2f 2X�0��

fX�0�fY �0�

�

fX�0�fY �0�

1− 2r2∑r

i=1∑r

j=1 F2Y�i��j�

�0�

2f 2y �0�

�

with � = P�X > 0� Y > 0�− 1r2

∑ri=1

∑rj=1 FX�i��j �

�0�FY�i��j��0�. Therefore, the asymptotic

efficiency of �̂′˜R

relative �̂′˜

is the square root of the ratio of the asymptotic

generalized variances, where the generalized variance is the determinant of thecovariance matrix. Hence, it is easy to show that

eff ��̂′˜R� �̂′

˜� =

1− �4P�X < 0� Y < 0�− 1�2

4(1− 2

r2

∑ri=1

∑rj=1 F

2X�i��j�

�0�)(

1− 2r2

∑ri=1

∑rj=1 F

2Y�i��j�

�0�)− 16�2

12

�

(4.1)

Using the results in Theorem 3.2, hence eff��̂′˜R� �̂′

˜� = �det�VV−1

R ��12 = ��1�2�

12 ≥ 1.

5. Numerical Comparison

The performance of the BVRSS sign test will be compared with the performanceof the BVSRS sign test, via the ratio of their noncentrality parameters for testing

Table 1Relative asymptotic efficiency of BVRSS sign test w.r.t. BVSRS sign test (negative

correlation values are in parentheses)

�1 = 0�4 and �2 = 0�9 or�1 = �2 �1 = 0�9 and �2 = 0�4

R � = 0 � = ±0�50 � = ±0�95 � = 0 � = ±�0�50 � = ±0�95

2 1.33 1.60 (1.10) 1.70 (1.03) 1.33 1.48 (1.13) 1.32 (1.04)3 1.60 2.22 (1.14) 2.34 (1.16) 1.60 1.98 (1.22) 1.68 (1.18)4 1.83 2.92 (1.17) 2.90 (1.41) 1.83 2.52 (1.29) 2.06 (1.43)5 2.03 3.72 (1.19) 3.35 (1.84) 2.03 3.15 (1.36) 2.51 (1.86)

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1082 Samawi et al.

Table 2The asymptotic relative efficiency of �̂′

˜Rrelative �̂′

˜R

r � = 0 � = ±0�50 � = ±0�95

2 1.33 1.33 1.333 1.60 1.60 1.654 1.82 1.84 2.025 2.03 2.10 2.50

H0 � � = 0 vs. H1 � � �= 0 in the case of underlying bivariate normal distribution of�X� Y �. In comparing the two tests we use the following configuration: set sizesr = 2� 3� 4, and 5, correlation coefficient � = 0�±0�5�±0�95, and �1 = �2, �1 = 0�4and �2 = 0�9. The asymptotic efficiency of �̂′

˜Rrelative to �̂′

˜is evaluated at the

different combinations of the values of r and �.Table 1 shows that BVRSS sign test is substantially more efficient, and hence

is more powerful, than BVSRS test for the bivariate location model. Clearly, theefficiency increases as the set size r increases for any given value of � = 0�±0�5,±0�95. It also seems that positive correlations lead to higher efficiency. This may bedue to the nature of bivariate ranked set sampling procedure. When the correlationis negative, ranking on one characteristic and then on the other characteristicmay associate the minima of the first characteristic with higher-order statistic ofthe second characteristic and not with its minima as hoped. Although, eventuallythe RSS for both characteristics are presented in the BVRSS but each orderstatistic of one characteristic will be reversely affected by the ranking of the othercharacteristic depending on the strength of the negative correlation between them,i.e., the structure of the BVRSS obtained when the correlation is positive differsfrom that when the correlation is negative. Moreover, the efficiency of the BVRSSsign test is higher when �1 = �2 than when �1 = 0�4 and �2 = 0�9 only for � > 0.From the table, again we see that when �1 = 0�4 and �2 = 0�9, the efficiency of theBVRSS sign test with respect to the BVSRS sign test increases as � increases from0 to 0.50 and then slightly decreases as � increases from 0.50 to 0.95.

Table 2 shows that using the sample medians of BVRSS for estimating thelocations of the bivariate model is also substantially more efficient than using thesample medians of BVSRS. The efficiency increases as the set size, r, increases forany given value of ��� and vice versa.

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