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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: omer@stat.ohio-state.edu
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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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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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.
Statist. Assoc., 1982, 77, 639646.
2. Boos, D. A test of asymmetry associated with the HodgesLehmann
estimator, J. Amer. Statist. Assoc., 1982, 77, 647651.
3. Koziol, J.A. Test for symmetry about unknown value based on
the empirical distribution function, Commun. Statist., Theory and
Methods, 1983, 12(24), 28232844.
4. Schuster, F.; Barker, R.C. Using the bootstrap in testing symmetry
versus asymmetry, Commun. Statist., Simulation, 1987, 16(1), 6984.
5. Randles, R.H.; Fligner, M.A.; Policello, G.E.; Wolfe, D.A. An
asymptotically distribution-free test for symmetry versus asymmetry,J. Amer. Statist. Assoc., 1980, 75, 168172.
6. O ztu rk, O . One and two-sample sign tests for ranked-set samples
with selective designs. Commun. Statist., Theory Methods, 1999, 28,
12311245.
2132 O ZTU RK
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8/22/2019 A nonparametric test for symmetry using RSS.pdf
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ORDER REPRINTS
7. O ztu rk, O .; Wolfe, D.A. Optimal allocation procedure in ranked-set
sampling for uni-modal and multi-modal distributions. Environmental
and Ecological Statistics, 2000, 7, 343356.
8. O ztu rk, O .; Wolfe, D.A. An improved ranked-set two-sample Mann
WhitneyWilcoxon test. Canad. J. Statist., 2000, 28, 123135.
9. Chen, Z. On ranked-set sample quantiles and their applications.
J. Statist. Plann. Infer., 2000, 83, 125135.
10. Chen, Z. The optimal ranked-set sampling scheme for inference on
population quantiles. Statistica Sinica, 2001, in print.
11. Kaur, A.; Patil, G.P.; Taillie, C. Ranked-set sample sign test under
unequal allocation. Penn State University, Center for Statistical
Ecology and Environmental Statistics, Department of Statistics.
Tech. Rep. 96-0692, 1996.12. Kaur, A.; Patil, G.P.; Taillie, C. Unequal allocation models for ranked-
set sampling with skew distributions. Biometrics, 1997, 53, 123130.
13. Kvam, P.H.; Samaniego, F.J. On the inadmissibility of empirical
averages as estimators in ranked-set sampling. J. Statist. Plann. Infer.,
1993, 36, 3955.
14. Stokes, S.L. Parametric ranked-set sampling, Ann. Inst. Statist. Math.,
1995, 47, 465482.
15. Bohn, L.L. A ranked-set sample signed-rank statistic. J. Nonpar.
Statist., 1998, 9, 295306.
16. Chen, Z.; Bai, Z. Optimal ranked-set sampling scheme for parametric
families. Sankhya, Series A., 2001, in print.
17. O ztu rk, O . Ranked-set sample inference under a symmetry restriction,
Tech. rep. 641, 1998, Ohio State University, Department of Statistics.
18. Stokes, S.L.; Sager, T.W. Characterization of a ranked-set sample with
application to estimating distribution functions, J. Amer. Statist.
Assoc., 1988, 83, 374381.
19. Fine, T. On the Hodges and Lehmann shift estimator in the two sample
problem, Ann. Math. Statist., 1966, 37, 18141818.
20. Parzen, E. On estimation of a probability density function and mode.
Ann. Math. Statist., 1962, 33, 10651076.
Received October 1999
Revised February 2001
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