A class of distribution-free tests for two-sample location problem is based on the

signs of most extreme observations in the sub-samples of sizes c and d from X

and Y samples respectively. The test statistics have been expressed in terms of

linear rank statistics. The asymptotic normality of the test statistics is established.

Asymptotic e�ciencies indicate that members of our class do well in comparison

with some already existing test statistics for light and medium tailed distribution

Kumari Savitha, .

Parameshwar V. Pandit, .

ePrints@Bangalore University

Reprint ISSN0973-9424I J OM SA EANTERNATIONAL OURNAL FATHEMATICAL CIENCESND NGINEERINGPPLICATIONS(IJMSEA)PUNE, INDIAlASCENT PUBLICATIONlwww.ascent-journals.com@ASCENTInternational J. of Math. Sci. & Engg. Appls. (IJMSEA)ISSN 0973-9424, Vol. 10 No. III (December, 2016), pp. 151-158ON DISTRIBUTION FREE TESTS FOR THE TWO-SAMPLELOCATION PROBLEM BASED ON SIGNS OF EXTREMEOBSERVATIONSPARAMESHWAR V. PANDIT1 AND SAVITHA KUMARI21 Department of Statistics, Bangalore University,Bangalore -560056, India2 Department of Statistics, SDM College, Ujire-574240, IndiaAbstractA class of distribution-free tests for two-sample location problem is based on thesigns of most extreme observations in the sub-samples of sizes c and d from Xand Y samples respectively. The test statistics have been expressed in terms oflinear rank statistics. The asymptotic normality of the test statistics is established.Asymptotic efficiencies indicate that members of our class do well in comparisonwith some already existing test statistics for light and medium tailed distributions.1. IntroductionThe two sample location problem is one of the fundamental problems encountered inStatistics. In many applications of statistics, two-sample problems arise in such a way asto lead naturally to the formulation of null hypothesis to the effect that the two samplescome from identical populations. There are many non-parametric tests available in−−−−−−−−−−−−−−−−−−−−−−−−−−−−Key Words : Two-sample location problem, Linear rank statistics sub-samples, Pitman effi-ciency.AMS Subject Classification : 62G10.c© http: //www.ascent-journals.com151152 PARAMESHWAR V. PANDIT & SAVITHA KUMARIliterature for the two sample location problem, their relative efficiency and suitabilitydepending on the nature of the (unknown) underlying distribution. Wilcoxon-Mann-Whitney W -test is a popular nonparametric test for this problem. Besides W-test anumber of distribution-free tests are available in the literature. Mathisen [4] proposeda test for this problem based on the number of observations in the X-sample not ex-ceeding the median of Y -sample. Moods median test (M) is particularly effective indetecting shift in location in the normal distribution. Gastwirths [3] L and H tests areeffective in detecting shifts in moderately heavy tailed distributions. The RS test dueto Hogg, Fisher and Randles [5] is effective in detecting shifts in distributions that areskewed. During the last decade or so, new classes of tests based on the so called sub-sample approach have been proposed for the above problem, notable among them beingDeshpande and Kochar [2], Stephenson and Ghosh [12], Shetty and Govindarajulu [10]and Shetty and Bhat [11] and Ahmed [1]. While Shetty and Govindarajulu [10] andShetty and Bhat [12] based their tests on sub sample medians which tend to emphasizethe centre of the underlying distributions, the other two are based on statistics involv-ing sub sample extreme with the object of gaining more information from the tails ofsampled distributions. The results of these papers demonstrate that the sub sampleapproach, applied selectively, does help to improve upon the efficiency performance ofthe tests in an overall sense. For example, Shetty and Govindarajulu [10] test performson one hand better than the Mann-Whitney test for heavy-tailed distributions, whileperforming better than the median test for light-tailed distributions on the other.Deshpande and Kochar [2] test, on the other hand, being sensitive to light tailed dis-tributions, performs substantially better than Mann-Whitney test for such underlyingdistributions and some what better for normal, while maintaining reasonable level ofefficiency under heavy tailed distributions. Stephenson and Ghosh [12] and Ahmed [1]tests are also relatively more sensitive than the Mann-Whitney test but less sensitivethan the Deshpande and Kochar [2] test to the light tailed distributions.In this paper, we propose a new class of the distribution-free test statistics which isthe convex combination of two U -statistics. Among the U -statistics involved in thecombination, one has the kernel based on subsample maxima and the other has thekernel based on sub sample minima. The distributional properties of the proposedclass is studied and the asymptotic relative efficiencies of few members of the classON DISTRIBUTION FREE TESTS FOR THE TWO-SAMPLE LOCATION... 153are investigated relative to few other statistics exist in the literature, particularly theoptimal member of the class proposed by Xie and Priebe [13].The SG test proposed by Shetty and Govindarajulu [10] based on subsample medi-ans takes care of two suspected outliners at extremes of both the samples. Deshpandeand Kochar [2] test is effective in detecting shift in distributions that are light tailed.Stephenson and Ghosh [12], U(c, d) and Shetty and Bhat [11] T (c, d) tests are few othertest procedures for this problem. In this section, we propose a class of distributionfree tests which are effective in detecting the shifts in distributions that are symmetric,medium and light tailed. The test statistics is proposed in section 2. An alternativeexpression for the class of test statistics is given in section 3. The distributional prop-erties of the proposed class of tests are presented in section 4. Section 5 is devoted tothe study of Pitman asymptotic relative efficiency and section 6 for some comments.2. Methods2.1. The Proposed Class of TestsSuppose X1, X2, · · · , Xm and Y1, Y2, · · · , Yn are independent random samples form con-tinuous distribution with c.d.f.’s F (x) and F (x − θ) respectively. We wish to testH0 : θ = 0 against H1 : θ > 0 with F (x) + F (−x) = 1.We propose a test based on U -statistic which is given by,V(c,d)(X1, X2, · · · , Xm, Y1, Y2, · · · , Yn)=1(mc)(nd)∑Ah[Xi1, Xi2, · · · , Xic;Yj1, Yj2, · · · , Yjd]where A denotes the sum over all(mc)(nd)combinations of X and Y sampleobservations and h(x1, · · · , xc; y1, · · · , yd) = h2(y1, · · · , yd)− h1(x1, · · · , xc).Hereh1(x1, · · · , xc) =1 if the largest in absolute value among(x1, · · · , xc) is positive0 otherwise154 PARAMESHWAR V. PANDIT & SAVITHA KUMARIandh1(y1, · · · , yd) =1 if the largest in absolute value among(y1, · · · , yd) is positive0 otherwise2.2. An Alternative Expression for V (c, d)The computational effort associated with the evaluation of V (c, d) is enormous whenthe sample sizes are large. However, it is possible to derive an alternative expression forV (c, d) as linear rank statistics.Suppose X|1|, · · · , X|m| are the observations arranged in the order of increasing absolutevalue. Consider distinct sub-samples of size c for which X\|k| is the largest in absolutevalue. For k ≥ c, we have (c−1) places in the sub samples that can be filled with (k−1)objects. X|1|, · · · , X|k−1|, each of which may appear at most once. Therefore, there are(k − 1c− 1)= bc(k), distinct sub-samples for whichh1(xi1, xi2, · · · , xic) =1, if X|k| > 00, otherwiseThe sum of (xi1, xi2, · · · , xic) over all distinct sub-samples of size c, 1 ≤ i1 ≤ i2 · · · ≤ic ≤ m, is equivalently given byV1 =m∑k=cbc(k)γk, where γk =1, if X|k| > 00, otherwise.On similar lines, we can prove that the sum of h2(yj1, yj2, · · · , yjd) over all distinctsub-samples of size d, 1 ≤ j1 ≤ j2 · · · ≤ jd ≤ n, is equivalently given byV2 =n∑k=dbd(k)ξk, where ξk =1, if Y|k| > 00, otherwise.Hence Y|1|, · · · , Y|n| are the observations arranged in increasing absolute value. Thenthe statistic V (c, d) can be written as(mc)(nd)V (c, d) =(mc)V2 −(nd)V1=(mc) n∑k=dbd(k)ξk −(nd) m∑k−cbc(k)γk.ON DISTRIBUTION FREE TESTS FOR THE TWO-SAMPLE LOCATION... 1553. Results3.1. Distributional Properties of V (c, d)The mean of V (c, d) is given byµ(θ) = E[V (c, d)] =∫ ∞−θ[F (y)− F (−y − θ)]d−1f(y)dy − 12> 0.Under H0, E[V (c, d)] = 0.However, under H1E[V (c, d)] =∫ ∞−θ[F (y)− F (−y − θ)]d−1f(y)dy − 12> 0, for d ≥ 2.Here, one can notice that V (c, d) is distribution free under H0. Since V (c, d) is a U -statistic with a square integrable kernel, the asymptotic normality of V (c, d) followsfrom Lehmann [6]. Under H0,√N V (c, d) is asymptotically normal with mean zero andvariance given byσ2c,d =c2ζ10λ+d2ζ011− λ,whereζ10 =14(2c− 1) , ζ01 =14(2d− 1) and λ = limN→∞mN, N = c+ d.3.2 Asymptotic Relative EfficiencyFor the sequence of Pitman alternatives θN = θ√N the efficacy of V (c, d) is given bye2v(d) = 4d2(d− 1)2{∫ ∞0[2F (y)− 1]d−2f2(y)dy}2/σ2c,d, d ≥ 2.When sub sample sizes are equal i.e, c = d = r, the efficacy of V (r) is given bye2v(r) = 16(r − 1)2(2r − 1){∫ ∞0[2F (y)− 1]r−2f2(y)dy}2/λ1− λ, r ≥ 2.Table 1, lists asymptotic relative efficiency of V (r) with respect to the two sample t testT for various continuous distributions for some values of r. Table 2, gives the ARE’sof V (r) with respect to Mann Whitney test M . The ARE’s of V (c, d) with respect toDeshpande and Kochar [2] Test L(c, d) for c = 1 and d = 2, 3, 4 for equal sample sizesare given in Table 3. In Table 4, the ARE’s of V (c, d) with respect to Stephenson andGhosh [12] test U(c, d) are presented.156 PARAMESHWAR V. PANDIT & SAVITHA KUMARITable 1: Asymptotic Relative Efficiency of V (r) relative to TDensity r = 2 r = 3 r = 4 CommentCauchy 0.3040 0.1792 0.1093 Max at r = 1Laplace 1.5000 1.1111 0.8750 Max at r = 1Logestic 1.0966 1.0281 0.9212 Max at r = 2Normal 0.9549 0.9774 0.9386 Max at r = 3Triangular 0.8889 0.9481 0.9752 Increasing in rParabolic 0.8640 1.0635 1.1836 Increasing in rUniform 1.0000 1.6667 2.3333 Increasing in rInv. Triangular 2.6667 6.4000 10.2857 Increasing in rTable 2 : Asymptotic Relative Efficiency of V (r) relative to MDensity r = 2 r = 3 r = 4Cauchy 0.1014 0.0597 0.0363Laplace 1.0000 0.7407 0.5833Logestic 1.0000 0.9375 0.8401Normal 1.0000 1.0236 0.9829Triangular 1.0000 1.0667 1.0971Parabolic 1.9812 2.4387 2.7141Uniform 4.6182 5.7731 8.0821Inv. Triangular 5.6558 13.5737 21.8148Table 3 : Asymptotic Relative Efficiency of V (1, d) relative to L(1, d)Density d = 2 d = 3 d = 4Laplace 1.1429 0.9524 0.8632Logestic 1.1428 1.2054 1.175Normal 1.1427 1.3158 1.3379Uniform 1.1428 2.2948 2.6437Table 4 : Asymptotic Relative Efficiency of V (1, d) relative to U(1, d)Density d = 2 d = 3 d = 4Cauchy 1.0000 0.6288 0.4808Laplace 1.0000 0.7407 0.7347Logestic 1.0000 1.0000 1.0000Normal 1.0000 1.0918 1.1389Triangular 1.0000 1.1378 1.2461Uniform 1.0000 1.7778 2.2500ON DISTRIBUTION FREE TESTS FOR THE TWO-SAMPLE LOCATION... 1574. Discussions1. It follows from Lehmann [6] that the test is consistent for testing H0 against H1.Since expected value of V (c, d) under H1 is greater than its expected value underH0 and the asymptotic distribution of the test statistic is normal.2. The proposed class of test statistic is constructed in such a way that the statisticis based on the signs of the observations that is the largest value in subsamples ofsizes c and d taken from X and Y samples.3. The performances of the members of our class are better than Mann-Whitneystatistics for light tailed distributions or the distributions with finite range.4. For medium tailed distribution, the members of our class are better for r > 2.5. For r = 2, our test is the best test statistics for light as well as heavy taileddistribution except for Cauchy distribution.6. It can be seen that ARE’s increase with increase in the sub sample size r. Forheavy tailed distributions, ARE’s decrease as r increases and for medium taileddistributions, ARE’s increase and then decrease as r increases.7. The performance of the members of our class is better as compared to Deshpandeand Kochar’s [2] test for both heavy and light tailed distributions.8. The performance of the members of our class is better as compared to Stephensonand Ghosh [12] test for light and medium tailed distributions.AcknowledgementThe second author would like to thank the University Grants Commission for its sup-port under FDP scheme.158 PARAMESHWAR V. PANDIT & SAVITHA KUMARIReferences[1] Ahmad I. A., A class of Mann-Whitney-Wilcoxon type statistics. The AmericanStatistician. 50 (1996), 324-327.[2] Deshpande J. V. and Kochar S. C., Some competitors of Wilcoxon-Mann Whit-ney test for the location alternatives. J. Indian Statist. Assoc. 19 (1982), 9-18.[3] Gastwirth J. L., Percentile modifications of two sample rank tests. J. Amer.Statist. Assoc, 60 (1965), 1127-1141.[4] Hoeffding W., A class of statistics with asymptotically normal distribution. Ann.Math. Statist., 19 (1948), 293-325.[5] Hogg R. V., Fisher D. M. and Randles R. H., A two-sample adaptive distribution-free test. J. Amer. Statist. Assoc., 70 (1975), 656-661.[6] Lehmann E. L., Consistency and unbiasedness of certain nonparametric tests.Ann. Math. Statist., 22 (1951), 165-179.[7] Mann H. B. and Whitney D. R., On a Test of whether one of the two randomvariables is stochastically larger than the other. Ann. Math. Statist, 18 (1947),50-60.[8] Mathisen H. C., A method of testing the hypothesis that two samples are fromthe same population. Ann. Math. Statist, 14 (1943), 188-94.[9] Randles R. H. and Wolfe D. A., Introduction to the Theory of Non-parametricStatistics. John Wiley and Sons, New York. (1979).[10] Shetty I. D. and Govindarajulu Z., A two-sample test for location. Commun.Statist.- Theor. Meth. 17(7) (1988), 2389-2401.[11] Shetty I. D. and Bhat S. V., A note on the generalization of Mathisens mediantest. Statistics and Prob. Letters, 19 (1994), 199-204.[12] Stephenson W. R. and Gosh M., Two sample non-parametric tests based onsubsamples. Commun. Statist. -Theor. Meth., 14 (1985), 1669-1684.[13] Xie J. and Priebe C. E., Generalizing the Mann-Whitney Wilcoxon Statistic.Non-parametric Statistics, 12 (2000), 661-682.

ON DISTRIBUTION FREE TESTS FOR THE TWO-SAMPLE

LOCATION PROBLEM BASED ON SIGNS OF EXTREME

OBSERVATIONS

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ON DISTRIBUTION FREE TESTS FOR THE TWO-SAMPLE LOCATION PROBLEM BASED ON SIGNS OF EXTREME OBSERVATIONS

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