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A NONPARAMETRIC TEST OF SYMMETRY VERSUS

ASYMMETRY FOR RANKED-SET SAMPLESmer ztrk a

a The Ohio State University, 1465 Mt. Vernon Avenue, Marion, OH, 43302, U.S.A.

Version of record first published: 15 Feb 2007.

To cite this article: mer ztrk (2001): A NONPARAMETRIC TEST OF SYMMETRY VERSUS ASYMMETRY FOR RANKED-SETSAMPLES, Communications in Statistics - Theory and Methods, 30:10, 2117-2133

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A NONPARAMETRIC TEST OF

SYMMETRY VERSUS ASYMMETRY

FOR RANKED-SET SAMPLES

O mer O ztu rk

The Ohio State University, 1465 Mt. Vernon Avenue,

Marion, OH 43302

E-mail: [email protected]

ABSTRACT

This paper introduces a nonparametric test of symmetry for

ranked-set samples to test the asymmetry of the underlying

distribution. The test statistic is constructed from the Crame r-

von Mises distance function which measures the distance

between two probability models. The null distribution of the

test statistic is established by constructing symmetric boot-

strap samples from a given ranked-set sample. It is shown

that the type I error probabilities are stable across all practical

symmetric distributions and the test has high power for asym-

metric distributions.

Key Words: Bootstrap distribution; Cramer-von Mises dis-

tance; Design; HodgesLehmann estimator; Median; Mode;

Unequal allocation

Mathematics Subject Classification Codes (1991): 62G30;

62H10

2117

Copyright & 2001 by Marcel Dekker, Inc. www.dekker.com

COMMUN. STATIST.THEORY METH., 30(10), 21172133 (2001)


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1. INTRODUCTION

Appropriate location and scale measures for a distribution depend

on its general structure. Either the mean or median can be used to define

the location of an underlying symmetric distribution, but it is not clear

how to define the appropriate location measure when the underlying

distribution is asymmetric. Similarly, the standard deviation makes sense

as a measure of spread (or scale) if the underlying distribution is symmetric,

but in an asymmetric setting, the researcher has to reflect more intensively

about what to use as a measure of spread for the underlying distribution.

Therefore, to select an appropriate statistical analysis, it is necessary to

check if the data set is coming from a symmetric distribution. There are

several available testing procedures to check the symmetry of the underlyingdistribution for a simple random sample (Antille, Kersting and Zucchini (1),

Boos (2), Koziol (3), Schuster and Barker (4), Randles, Fligner, Policello

and Wolfe (5)). The finite sample null distributions of all these testing

procedures, with the exception of Schuster and Barker (4) test, depend on

the underlying probability models. Schuster and Barker use the bootstrap

distribution to get around this problem.

In many scientific investigations, where the actual measurement of

the observations is difficult relative to a complete or partial ranking of all

units in a set, ranked-set sampling provides improved statistical inference.

The data collection procedure in a ranked-set sampling involves selecting m2

units at random from an infinite population F. These units are divided

into m sets, each having m units. The units are then judgment ranked by

some criterion that does not require actual measurements. The ith judgment

order statistic, Xi1, from the ith set is quantified for i 1, . . . , m.The observations, Xi1, i 1, . . . , m, constitute a cycle and the process canbe repeated for n cycles to have n m quantified observations, Xij,i 1, . . . , m; j 1, . . . , n. If the judgment ranking is perfect, the ith

judgment order statistic is the same as the ith order statistic for each of the

n cycles. In this case we use the usual order statistic notation, Xij, to denotethe ith order statistic from the jth cycle, i 1, . . . , m, j 1, . . . , n. Unlessstated otherwise, we assume perfect judgment ranking throughout the paper.

Note that the ranked-set sample described above is balanced, namely

each of the m order statistics is quantified an equal number of times. Recent

research has shown that unequal allocation in a ranked-set sample improves

the efficiency of a balanced ranked-set sample (O

zt uurk (6); O

zt uurkand Wolfe (7,8), Chen (9,10), Kaur, Patil and Taillie (11,12)). In the

same spirit but in a different context, Kvam and Samaniego (13) and

Stokes (14) used unequal weighting in L-estimators. Thus, we consider

unequal allocation to cover several optimal allocation procedures.

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For a fixed t 2 f1, . . . , mg, let D fd1, . . . , dtg be a set of integersthat determines the judgment ranks of the quantified observations. For

example, if D f1, 5g and the set size m is 5, we quantify two extremesin each cycle. This definition is general enough to cover all practical alloca-

tion procedures. For example, O zt uurk and Wolfe (7,8) showed that ranked-

set samples that quantify observations at the modes of the underlying

distribution improve the Pitman efficiency for one- and two-sample

nonparametric testing procedures. Thus, the optimal design for unimodal

distributions quantifies the median in each set which will be covered

by D fm=2g or D fm=2 1g ifm is even and D fm 1=2g otherwise.Other practical allocation procedures can be considered in a similar fashion.

The structure of symmetry on F not only gives a meaning to the

location and scale parameters, but also improves the efficiency of the

parametric and nonparametric statistical inferences. For example, Bohns

(15) the ranked-set sample signed ranked test has higher Pitman efficiency

than its simple random sample analog. On the other hand, use of the signed

rank test requires a symmetric underlying distribution. Recently, Kaur,

Patil and Taillie (12) and Chen and Bai (16) showed that for a symmetric

distribution, the optimal design for estimation of the population mean

quantifies only the middle observations. For symmetric distributions,

optimal designs for the one and two-sample nonparametric tests quantify

the observations at the modes of the underlying distribution, O zt uurk and

Wolfe (7,8). In a recent work, O zt uurk (17) introduced an estimator for

the symmetric distribution functions that improves the efficiency of the

ranked-set sample cdf (cumulative distribution function) estimator ofStokes and Sager (18). All these facts combine to demonstrate that there

is a need to develop a test of symmetry for the ranked-set sampling scheme

similar to what is available for simple random sampling. In this paper,

we propose to develop such a test of symmetry for the underlying distribu-

tion when the researcher collects data according to a ranked-set sampling

protocol.

Section 2 introduces the test statistic. It is shown that the test statistic

arises naturally from the Crame r-von Mises distance function. Section 3

considers the bootstrap null distribution of the test statistic. It is evident

that the bootstrap null distribution must be established under the symmetry

restriction. It is shown that a symmetric empirical cdf exists for any ranked-

set sample. The performance of the test is studied in Section 4 via a small

simulation study. It is shown that the null distribution of the test statistic

is stable across all practical symmetric distributions and the test has good

power for asymmetric distributions. We provide concluding remarks in

Section 5. All proofs are given in the Appendix.

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2. TEST OF SYMMETRY

Let O be the space of all continuous symmetric distribution about ,i.e., O fFx : Fx 1 F2 x, for every xg. For a fixed, but arbi-trary design D, let Xdij, i 1, . . . , t; j 1, . . . , n, be a ranked-set samplefrom a population having the cumulative distribution function (cdf ) F.

Based on this sample, we consider devising a nonparametric test for

H0: Fx 2O against H1: Fx 62 O

where is the unknown point of symmetry. Any sensible test for this

hypothesis must estimate and select a suitable test statistic. For a given

design D, let

FD, ny 1

tn

Xti1

Xnj1

Xdij y

be the Stokes and Sager (18) cdf estimator of F, where a 1,0 asa , > 0.

Definition 1. The design D is called symmetric if di dm1i m 1, fori 1, . . . , t.

We note that FD, ny

is not necessarily an unbiased estimator for F

if the design D does not correspond to the standard ranked-set sampledesign D SRSS f1, . . . , mg. On the other hand, it is an unbiased esti-mator for

Pti1 Fdisry=t, which is a symmetric distribution if the under-

lying probability model F and the design D are symmetric. Thus, even

though FD, nt is a biased estimator for Ft, it is still possible to useFD, nt to devise a test of symmetry as long as D is a symmetric design.Therefore, we restrict our attention to the symmetric designs D in the

remainder of this paper. Let

W Z1

1fFD, ny FD, n2y 1g2dy:

A functional form of W can be written as

W Z1

1

1

t

Xtj1

Fdjy Fm1dj2y 1( )2

dy

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where Fdi is the cdf of the dith order statistic. If F 2 O, then FdjyFm1dj2y 1 0 for all y and j 1, . . . , m. Thus, under the nullhypothesis for large values of n we expect W to be close to zero andwe reject the null hypothesis for large values of it.

From calculations similar to those of Fine (19) W can be written as

W 1t2n2

Xti1

Xnj1

Xtk1

Xnr1

jXdij Xdkrj

1t2n2

Xti1

Xnj1

Xtk1

Xnr1

jXdij Xdkr 2j: 1

Note that if the underlying distribution F is symmetrical about and the

design D is symmetric, then

Xti1

Xtk1

jXdij Xdkr 2jdXti1

Xtk1

jXdij Xdkrj,

where d stands for the equality in distribution. We propose to useBoos (2) test statistic to asses the symmetry of the underlying distribution

F, namely,

Tn n 1 Pt

i1 Pnj1 Ptk1 Pnr1 jXdij Xdkr 2jPti1

Pnj1

Ptk1

Pnr1 jXdij Xdkrj

( ), 2

where is a suitable estimator for the point of symmetry. We note that

the denumerator of the ratio in Eq. (2) is the nt 1=nt times Ginismean difference which estimates the dispersion of the underlying distribu-

tion. Thus, W is standardized with respect to the dispersion of F.If is a consistent estimator for the point of symmetry , under the

null hypothesis the second term in the curly brackets in Eq. (2) is,

in distribution, equal to one. Thus, we expect that Tn will be close tozero under the symmetry assumption and we reject the null hypothesis

for large values of Tn. The problem with this approach is that the nulldistribution of the test statistic Tn

^, as in the simple random samplingBoos (2), depends on the underlying probability model. Boos has shown

that the null limiting distribution of Tn in a simple random sample isan infinite sum of the weighted chi-square random variables. On the other

hand, the null distribution of Tn appears to be stable across many

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practical unimodal symmetric distributions if it is evaluated at the logistic

model. The limiting null distribution of Tn in a ranked-set sample willalso be a function of an infinite sum of independent chi-square random

variables. On the other hand, it will be much more complicated than

its simple random sample counterparts since it involves order statistics.

Thus, it will not have much use in practical applications. In order to est-

ablish the null distribution of Tn, in Section 3, we study the bootstrap nulldistribution.

3. BOOTSTRAP DISTRIBUTION OF Tnhh

A close investigation of Eq. (1) shows that W is a convex functionof. Thus, there exists a unique minimizer. Let

HL arg min2R

W:

O zt uurk (17) showed that HL is the median of all Walsh averages XdijXdkr=2, i, k 1, . . . , t; j, r 1, . . . , n. For the standard ranked-set sampleD SRSS the asymptotic variance of ffiffiffinp HL is

2 1

6m

m

1

fRf2

y

dy

g2:

For a general design D, the variance offfiffiffi

np

HL is given in O zt uurk (7).

We construct our test statistic by replacing with HL in Eq. (2) and

reject the null hypothesis H0 in favor ofH1 for large values ofTn TnHL.Our test rejects the null hypothesis of symmetry if and only if

TnHL TnFD, n, Xd11, . . . , Xdtn, HL ! t,

where

PFfTnHL ! tjH0 is trueg t, F

is the probability of type I error for the proposed test at t and F.Apparently, the type I error of the test is a function of the unknown under-

lying distribution F. Thus, the test is not distribution free. We approximate

t, F by using the bootstrap distribution of TnHL.

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The bootstrap is a nonparametric estimation procedure for the distri-

bution of some functional, say M MF, X1, . . . , Xn. An algorithm for thebootstrap distribution of M is given as follows:

For b 1 to BI Sample Xb1 , . . . , X

bn with replacement from the empirical CDF Fn

based on the original sample X1, . . . , Xn.

II Compute Mb MFn, Xb1 , . . . , Xbn :Then the bootstrap estimate of the distribution of M is the empirical dis-

tribution of Mi, i 1, . . . , B.It is clear that the bootstrap distribution of TnHL must be esta-

blished by utilizing the symmetry restriction. For a finite sample size n,FD, n is not symmetric even if the ranked-set sample, Xdij, 1, . . . , t;j 1, . . . , n, is generated from a symmetric distribution F. Thus, wecannot sample our bootstrap sample from FD, n since the bootstrap estimate

t, FD, n will not estimate the type I error probabilities correctly. In orderto get a good approximation for t, F, FD, n must be replaced with asuitable symmetric distribution. In the following theorem, we show that a

symmetric distribution exists for any symmetric design D and sample size n.

Theorem 1. Let Xdij, i 1, . . . , t; j 1, . . . , n, be a ranked-set samplefrom a symmetric distribution F 2 O. Then GD, ny, HL fFD, nyFD, n 2HL y 1g=2 is the closest symmetric distribution in O toFD, ny, where the closeness is measured with respect to the Cramer-vonMises distance function; i.e,

WHL Z1

1fFD, ny GD, ny, HLg2dy

minZ1

1FD, ny Gy2dy; G 2 O

:

Theorem 1 allows us to select a bootstrap sample from a symmetric

distribution which is the closest distribution to FD, n in Crame r-von Mises

distance. In order to construct a bootstrap sample under the null hypothesis,

we first estimate with HL. Then, we sample nt observations with replace-

ment from 2tn observations, Xdij ^HL, j 1, . . . , n; i 1, . . . , t. Thisbootstrap sample will be a realization from the distribution GD, ny, HL

and will be called symmetric bootstrap sample throughout the remainder

of the paper. We use symmetric bootstrap samples to construct our test of

symmetry.

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For b 1, . . . , B, let Tbn Tbn GD, n, Xb1 , . . . , Xbnt, bHL be the value ofTnHL obtained from the bth symmetric bootstrap sample. Then theempirical cdf of Tbn , b 1, . . . , B, estimates the distribution of TnHL.Let TTn TnFD, n, Xd11, . . . , Xdtn, HL be the value of TnHL obtainedfrom the original sample. Then,

B TTn XBb1

Tbn TTn=B

is the p-value of the test, which can be used to reject the null hypothesis

for its small values.

As suggested in Schuster and Barker (4), for bootstrap samples werecommend using a nearby continuous symmetric distribution instead of a

discrete GD, n. Let

GD, ny 1

nt

Xnti1

Ky Yi

a

,

where Y1, . . . , Ynt is a symmetric bootstrap sample from GD, n, K is the

cdf of the uniform distribution over 1, 1, and positive constant a isthe smoothing parameter. A good choice of a can be considered from

the optimal smoothing parameter which makes the derivative of GD, nya good estimate of the derivative of F

Dy

Pt

i1F

diy

=t. From

Lemma 4A of Parzen (20), the optimal smoothing parameter of this density

estimation is

aopt

2=52 f

Rk2y dyg1=5

fRf002Dy dyg1=5n1=5 , 3where ky is the kernel density, 2 is the second moment of thekernel density, and f00Dy is the second derivative of

Pti1 fdiy=t.

Unfortunately, aopt is not distribution free since it is a function of the under-

lying distribution F. For a simple random sample, Boos (2) has shown that

the null distribution of TnHL does not vary much across all practical

unimodal distributions if it is evaluated at the logistic distribution. Thus,we expect and show in a simulation study that the smoothing parameter

evaluated at the logistic distribution will perform sufficiently well across all

practical distributions. Let Hy be the logistic distribution with mean zeroand scale 1.

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Lemma 1. Let Fy Hy = be the logistic distribution with mean and scale . For a given design D, uniform kernel density over 1, 1 and a

perfect ranking, the optimal smoothing parameter for fDy is

aopt 1:353=5

c1=5D n

1=5,

where

cD Z1

t Xt

i

1

mm 1di

1

udi11 umdiAdiu

( )2u1 u du,

Adiu d2i m 12di 1u m 2m 1u2:

For an easy reference, Table 1 presents the values ofcD for selected practical

designs and set sizes.

In order to estimate , we recommend using interquartile range (IQR)

since it will be less affected by the extreme observations. For the logistic

distribution, the estimate of can be expressed as

S IQRD

,

where D is given in Table 1 and suggested for the unbiasedness of the

estimator at the logistic distribution and ranked-set sample design D.We note that one can use some other symmetric kernel densities in

GD, ny such as logistic or normal. On the other hand, the selection of theuniform kernel density over 1, 1 makes the sampling easy since GD, nycan be expressed as the convolution of GD, ny with the cdf of the uniform

Table 1. The Coefficient cD and D for Selected Set Sizes and Designs at the

Logistic Distribution Under the Perfect Judgment Ranking

m D D cD m D D cD

2 {1, 2} 2.197 0.024 4 {1, 2, 3, 4} 2.197 0.024

3 {1, 3} 2.748 0.009 5 {1, 5} 3.812 0.065

3 {2} 1.449 0.166 5 {2, 4} 1.705 0.0693 {1, 2, 3} 2.197 0.024 5 {2, 3, 4} 1.489 0.143

4 {1, 4} 3.334 0.033 5 {3} 1.156 0.494

4 {2, 3} 1.449 0.166 5 {1, 2, 3, 4, 5} 2.197 0.024

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distribution over a, a. Thus, we can sample a symmetric bootstrap samplefrom GD, ny by observing that Z Y aU is a realization from GD, ny,where Y and U are independent random variables, Yhas cdfGD, ny, and Uhas uniform cdf over 1, 1.

4. SIMULATION RESULTS

We must address two basic questions about the proposed testing

procedure: (1) How good is the type I probability approximation across

all practical distributions? and (2) How powerful is the test to detect

asymmetry?Table 2 contains the type I error probability estimates of the proposed

test. The simulation study is done with IMSL random generator for several

symmetric distributions. The distributions include normal, double exponen-

tial, Cauchy, logistic, uniform, beta(0.9, 0.9) and ti, i 2, 3, where ti is theStudents t-distribution with the degrees of freedom i. We considered sample

sizes n 10, 20, 30 and several symmetric designs. In the estimation of thetype I error probabilities, we generated 100 different samples for each

sample size, design and distribution combinations. For each one of these

100 samples, we generated 5000 symmetric bootstrap samples from GD, nyand calculated the test statistic. Based on these 5000 Tbn , b 1, . . . , B, weestimated the critical value for a five-percent test, i.e. ti : fP

Bb1 Tbn ! ti =

Bg

0:05, i

1, . . . , 100. The bootstrap estimate of the critical value of a

five-percent test is taken as the average of these 100 ti , i 1, . . . , 100,tt P100i1 ti=100.

In order to see how good tt estimates the true critical value, we gene-rated 10 000 independent samples for sample size, design and distribution

combinations. Let

Stt P10000

i1 f TTinHL ttg10 000

,

where TTinHL is the test statistic evaluated at an independent sample.Note that if tt is a good estimate for the critical value of a five-percenttest, we expect that Stts in Table 2 are approximately 0.05 for symmetricdistributions. The results in Table 2 are typical estimates of type I errorprobabilities for a five-percent test. There is not a big difference between the

nominal size and the true size of the test across all practical distributions

listed in Table 2. On the other hand, it is important to notice that the

designs that suggest a strong bi-modality for the generated ranked-set

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Table 2. Estimates of the Type I Error Probabilities for Several Symmetric

Distributions: Logistic(L), Normal(N), Double Exponential (DE), Cauchy(C),

Uniform(U), Students T-distribution with Degrees of Freedom i (ti), Beta

Distributions with Parameters 0.9 and 0.9 (B)

m D n L N DE C U t2 t3 B

2 {1, 2} 10 0.053 0.045 0.069 0.040 0.076 0.055 0.056 0.078

20 0.049 0.050 0.057 0.039 0.079 0.055 0.062 0.100

30 0.051 0.048 0.063 0.038 0.089 0.058 0.048 0.101

3 {1, 3} 10 0.036 0.031 0.052 0.041 0.048 0.059 0.049 0.055

20 0.040 0.031 0.059 0.041 0.052 0.058 0.051 0.058

30 0.038 0.032 0.057 0.033 0.061 0.052 0.057 0.068

3 {2} 10 0.045 0.050 0.063 0.069 0.052 0.052 0.055 0.05820 0.048 0.049 0.067 0.071 0.059 0.054 0.051 0.057

30 0.051 0.049 0.054 0.047 0.064 0.049 0.055 0.065

3 {1, 2, 3 } 10 0.052 0.047 0.063 0.037 0.063 0.065 0.058 0.071

20 0.052 0.051 0.057 0.033 0.071 0.061 0.056 0.080

30 0.052 0.049 0.064 0.034 0.073 0.059 0.057 0.083

4 {1, 4} 10 0.015 0.008 0.028 0.031 0.021 0.043 0.030 0.025

20 0.016 0.008 0.032 0.031 0.022 0.047 0.035 0.029

30 0.016 0.007 0.035 0.032 0.026 0.047 0.030 0.026

4 {2, 3} 10 0.048 0.048 0.057 0.061 0.054 0.056 0.054 0.059

20 0.052 0.051 0.060 0.052 0.055 0.058 0.055 0.057

30 0.048 0.049 0.057 0.050 0.060 0.054 0.052 0.065

4 {1, 2 , 3, 4 } 10 0.049 0.044 0.066 0.038 0.057 0.053 0.057 0.059

20 0.051 0.047 0.060 0.029 0.064 0.054 0.059 0.07030 0.050 0.042 0.059 0.033 0.062 0.052 0.051 0.074

5 {1, 5} 10 0.004 0.000 0.014 0.026 0.007 0.030 0.019 0.009

20 0.003 0.000 0.014 0.030 0.008 0.036 0.023 0.008

30 0.003 0.000 0.016 0.028 0.007 0.034 0.022 0.007

5 {2, 4} 10 0.042 0.043 0.055 0.056 0.046 0.054 0.051 0.048

20 0.044 0.043 0.051 0.056 0.048 0.050 0.044 0.053

30 0.049 0.043 0.055 0.050 0.057 0.054 0.049 0.057

5 {3} 10 0.051 0.050 0.060 0.053 0.046 0.055 0.054 0.049

20 0.050 0.047 0.059 0.053 0.051 0.049 0.049 0.052

30 0.046 0.051 0.056 0.060 0.058 0.051 0.051 0.056

5 {2, 3, 4 } 10 0.047 0.047 0.056 0.059 0.051 0.053 0.057 0.054

20 0.047 0.047 0.061 0.059 0.055 0.057 0.052 0.060

30 0.052 0.047 0.055 0.044 0.056 0.055 0.053 0.065

5 {1, 2 , 3, 4 , 5 } 10 0.044 0.041 0.055 0.027 0.047 0.050 0.054 0.054

20 0.042 0.040 0.050 0.025 0.048 0.046 0.051 0.057

30 0.044 0.040 0.048 0.026 0.051 0.048 0.049 0.055

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Table 3. Empirical Power of the Bootstrap Test of Symmetry for Several

Asymmetric Distributions: Chi Square Distribution with Degrees of Freedom i

(Chi-i), Lognormal Distribution (LN), Exponential Distribution (E), Weibull

Distribution with Shape Parameter 0.5 and Scale Parameter 3 ( W(0.5, 3))

m D n Chi-1 Chi-2 LN E W(0.5, 3)

2 {1, 2} 10 0.891 0.646 0.750 0.653 0.871

20 0.998 0.944 0.975 0.950 0.998

30 1.000 0.994 0.999 0.995 1.000

3 {1, 3} 10 0.954 0.754 0.813 0.776 0.941

20 1.000 0.984 0.995 0.985 1.000

30 1.000 1.000 1.000 1.000 1.000

3 {2} 10 0.333 0.167 0.214 0.169 0.34620 0.702 0.360 0.493 0.389 0.745

30 0.887 0.537 0.690 0.580 0.951

3 {1, 2, 3} 10 0.987 0.857 0.913 0.870 0.984

20 1.000 0.996 0.999 0.997 1.000

30 1.000 1.000 1.000 1.000 1.000

4 {1, 4} 10 0.964 0.745 0.857 0.797 0.945

20 1.000 0.993 0.998 0.996 1.000

30 1.000 1.000 1.000 1.000 1.000

4 {2, 3} 10 0.727 0.367 0.485 0.387 0.790

20 0.971 0.725 0.847 0.744 0.993

30 0.999 0.907 0.969 0.913 1.000

4 {1, 2, 3, 4} 10 0.999 0.963 0.982 0.962 0.999

20 1.000 1.000 1.000 1.000 1.00030 1.000 1.000 1.000 1.000 1.000

5 {1, 5} 10 0.957 0.711 0.859 0.785 0.945

20 1.000 0.989 0.999 0.996 1.000

30 1.000 1.000 1.000 1.000 1.000

5 {2, 4} 10 0.824 0.464 0.595 0.481 0.877

20 0.993 0.828 0.920 0.839 0.999

30 1.000 0.957 0.987 0.962 1.000

5 {3} 10 0.262 0.125 0.156 0.129 0.269

20 0.578 0.270 0.367 0.274 0.702

30 0.802 0.417 0.548 0.427 0.894

5 {2, 3, 4} 10 0.906 0.599 0.741 0.614 0.960

20 0.999 0.914 0.975 0.919 1.000

30 1.000 0.988 0.998 0.990 1.000

5 {1, 2, 3, 4, 5} 10 1.000 0.991 0.997 0.991 1.000

20 1.000 1.000 1.000 1.000 1.000

30 1.000 1.000 1.000 1.000 1.000

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sample underestimate the size of the test. For example, design D f1, 5gwith m 5 underestimates the type I error probabilities.

For asymmetric distributions, the critical value tt still estimates thecritical value of a five-percent test since it is calculated based on symmetric

bootstrap samples from GD, ny. Thus, stt yields the power of the test ifthe independent ranked-set samples are generated from an asymmetric dis-

tribution. The values of empirical power s tt are presented in Table 3 forseveral asymmetric distributions, selected designs and sample sizes. The

distributions include lognormal, exponential, Weibull (0.5, 3) and the chi

square distribution with degrees of freedom 1 and 2. Table 3 shows that

TnHL has good power for standard ranked-set samples (D f1, . . . , mg)and the designs that quantify extreme order statistics in each cycle. For

example, the designs D f1, 4g and D f1, 5g have higher powers thanthe designs D f2, 3g and D f2, 4g when m 4 and m 5, respectively.There is a loss of power for the central designs that quantify only the middle

order statistics as expected. This can be explained from the fact that asym-

metry for unimodal distribution can be expressed in the form of a right-

skewness or left-skewness. Thus, the central designs (such as D f3g withm 5) ignore the largest order statistic which contributes to the skewness(asymmetry) and result a loss in the power of the test.

5. CONCLUDING REMARKS

This paper establishes a test of symmetry for ranked-set samples to

assess the asymmetry of the underlying distribution. The null distribution of

the test is constructed by using the symmetric bootstrap samples. It appears

in Table 2 that the null distribution is stable across all practical distributions

and the sample sizes as small as n 10. A simulation study indicates thatthe test has high power for asymmetric distributions.

APPENDIX

Proof Theorem 1: Let F be an arbitrary distribution in O and HFD, n, Fbe the Cramer-von Mises distance measure between FD, n and F. For the

proof of the theorem it is sufficient to show that

min1


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We now consider

HFD, n, F Z

fFD, ny Fyg2dy Z

fFD, ny GD, n,yg2dy

2Z

fFD, ny GD, n,ygfGD, n,y Fyg dy

Z

fGD, n,y Fyg2dy: 4

Note that both F: and GD, n, : are in O. Let

U,y FD, ny GD, n,y FD, ny FD, n2y 12

:

It is clear that U,y is symmetric around and U,y U, 2y. Thecross-product term in Eq. (4) can be expressed as

ZfFD, ny GD, n,ygfGD, n,y Fyg dy

Z1

1U,yGD, n,y dy

Z11

U,yFy dy:

We now show that this cross-product term is equal to zero. We start with the

first integral

Z11

U,yGD, n,y dy Z

1U,yGD, n,y dy

Z1

U,yGD, n,y dy

Z1

U, 2 zGD, n, 2 z dz

Z1

U,yGD, n,y dy

Z1

U, zf1 GD, n, zg dz

Z1

U,yGD, n,y dy

Z1

U, z dz: 5

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ORDER REPRINTS

Since F 2 O, it follows that

Z11

U,yFy dy Z1

U,y dy:

By combining Eqs (5) and (6), we observe that the cross-product term is

zero. Therefore, we have shown that

minF2O

HFD, n, F W:

The remaining part of the proof of the theorem follows from Lemma 1 of

O zt uurk (17).

Proof of Lemma 1: For the uniform kernel over 1, 1, 2 1=3 andRk2ydy 1=2. Without loss of generality we assume 0. Then, we

need to evaluate the second derivative of

fDy 1

t

Xti1

hdiy=:

Using the fact that, for logistic distribution, fy Fyf1 Fyg, we canexpress h

d

iy=

as

hdiy= mm 1di 1

Hdi1y=f1 Hy=gmdihy=

m m 1di 1

Hdiy=f1 Hy=gmdi1:

After some algebraic manipulations, we have

h00diy= m

2m 1di 1

Hdi1y=f1 Hy=gmdiAdifHy=ghy=,

where

Aau a2 m 12a 1u m 2m 1u2:

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ORDER REPRINTS

Note that

CD Z

f002Dy dy Z

1

t

Xti1

h00diy=2

dy

Z

m

2t

Xti1

m

2m 1di 1

Hdi1y=:

f1 Hy=gmdiAdifHy=gh1=2y=2

hy= dy:

Let Hy= u. Then we have

CD 13

Zmt

Xt

i1

m 1di 1

udi11 umdiAdiuu1=21 u1=2

2

du cD3

,

which completes the proof.

ACKNOWLEDGMENT

The author thanks to the anonymous referee for his/her helpful com-

ments and suggestions.

REFERENCES

1. Antille, A.; Kersting, G.; Zucchini, W. Testing symmetry, J. Amer.

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7. O ztu rk, O .; Wolfe, D.A. Optimal allocation procedure in ranked-set

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Received October 1999

Revised February 2001

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