Simultaneously Testing for Location and Scale Parameters of ...1990), majority depth (Singh 1991) and projection depth (Donoho & Gasko 1992). The details about notion of data depth,

Revista Colombiana de EstadsticaJuly 2019, Volume 42, Issue 2, pp. 185 to 208

DOI: http://dx.doi.org/10.15446/rce.v42n2.70815

Simultaneously Testing for Location and ScaleParameters of Two Multivariate Distributions

Prueba simultánea de ubicación y parámetros de escala de dosdistribuciones multivariables

Atul Chavana, Digambar Shirkeb

Department of Statistics, Shivaji University, Kolhapur, India

Abstract

In this article, we propose nonparametric tests for simultaneously testingequality of location and scale parameters of two multivariate distributionsby using nonparametric combination theory. Our approach is to combinethe data depth based location and scale tests using combining function toconstruct a new data depth based test for testing both location and scale pa-rameters. Based on this approach, we have proposed several tests. Fisher’spermutation principle is used to obtain p-values of the proposed tests. Per-formance of proposed tests have been evaluated in terms of empirical powerfor symmetric and skewed multivariate distributions and compared to theexisting test based on data depth. The proposed tests are also applied to areal-life data set for illustrative purpose.

Key words: Combining function; Data depth; Permutation test; Two-sampletest.

Resumen

En este artículo, proponemos pruebas no paramétricas para probar si-multáneamente la igualdad de ubicación y los parámetros de escala de dosdistribuciones multivariantes mediante la teoría de combinaciones no para-métricas. Nuestro enfoque es combinar las pruebas de escala y ubicaciónbasadas en la profundidad de los datos utilizando la función de combinaciónpara construir una nueva prueba basada en la profundidad de los datos paraprobar los parámetros de ubicación y escala. Con base en este enfoque,hemos propuesto varias pruebas. El principio de permutación de Fisher seusa para obtener valores p de las pruebas propuestas. El rendimiento de laspruebas propuestas se ha evaluado en términos de potencia empírica para

aPhD Student. E-mail: [email protected]. E-mail: [email protected]

185

186 Atul Chavan & Digambar Shirke

distribuciones multivariadas simétricas y asimétricas y se comparó con laprueba existente basada en la profundidad de los datos. Las pruebas prop-uestas también se aplican a un conjunto de datos de la vida real con finesilustrativos.

Palabras clave: Función de combinación; Profundidad de datos; Prueba depermutación; Prueba de dos muestras.

1. Introduction

In real life, we come across with many ambiguous situations, where we haveto take a decision whether the location and scale parameters of two multivariatedistributions are equal or not. In such situations, testing of hypothesis is used toaddress such problems. Various multivariate two-sample tests for simultaneouslytesting location and scale parameters are available in the statistical literature. Butmost of these tests are based on the assumptions about underlying probabilitydistributions, particularly assumption of multivariate normality. In reality, suchassumption may not be satisfied and hence, we have to go for nonparametricmultivariate two-sample testing procedures.

Based on notion of data depth, many nonparametric tests for multivariatelocations and scales are available in literature. The word depth was first coinedby Tukey (1975) for picturing data. Many data depth functions were proposedin literature for capturing different probabilistic properties of multivariate data.Among all of them, more popular choices of data depth functions are halfspacedepth (Tukey 1975), Mahalanobis depth (Mahalanobis 1936), simplicial depth (Liu1990), majority depth (Singh 1991) and projection depth (Donoho & Gasko 1992).The details about notion of data depth, data depth functions, depth versus depthplot (DD-plot), inference using data depth are provided by Liu, Parelius & Singh(1999). Zuo & Serfling (2000) described four desirable properties for the depthfunctions.

Liu & Singh (1993) proposed two-sample rank tests based on data depth forsimultaneously testing shift in the location and change in the scale parameters.Rousson (2002) has proposed tests for testing both location and scale differencesbetween two multivariate distributions. Li & Liu (2004) have proposed two depth-based nonparametric tests viz. T -based test and M -based test for multivariatelocation difference. These tests for locations are based on the depth versus depthplot (DD-plot) introduced by Liu et al. (1999). Dovoedo & Chakraborti (2015)have reported an extensive simulation study to evaluate the performance of T -based and M -based tests for well-known family of multivariate skewed as well assymmetric distributions and compared the performance of these tests for four pop-ular affine-invariant depth functions. Chavan & Shirke (2016) have proposed twononparametric tests for testing equality of location parameters of two multivariatedistributions. These tests are extensions of the M -based test introduced by Li &Liu (2004). Li, Ban & Santiago (2011) have proposed two-sample nonparametrictests for comparing species assemblages. These tests can be considered as a natu-ral generalization of the Kolmogorov-Smirnov (KS) and Cramer-Von Mises (CM)

Revista Colombiana de Estadstica 42 (2019) 185–208

Simultaneously Testing for Location and Scale Parameters 187

tests. Chenouri & Small (2012) proposed a family of nonparametric multivariatemultisample tests based on depth rankings. Nonparametric tests for scale differ-ences between two samples have been provided by Li & Liu (2016). Many of thesetests use Fisher’s permutation tests to calculate the p-value.

In literature, for a univariate data, many nonparametric tests are proposedby combining the permutation-based location test and scale test for simultane-ously testing location and scale parameters. Based on squared ranks and squaredcontrary-ranks, Cucconi (1968) proposed a test for the location-scale problem.Lepage (1971) proposed a test which is the combination of the Wilcoxon test forlocation and the Ansari-Bradley test for scale. Podgor & Gastwirth (1994) deviseda test statistic by taking a quadratic combination of the rank test for location anda rank test for scale. Neuhäuser (2000) modified Lepage L-test by replacing theWilcoxon test for location with a location test proposed by Baumgartner, Weiß &Schindler (1998). Also, Murakami (2007) proposed a modification of the Lepagetest. More recently, Park (2015) proposed several nonparametric simultaneous testprocedures for location and scale parameters by using combining function. To thebest of our knowledge, it appears that relatively less literature has been reportedon combining location and scale tests for multivariate data

Sometimes assumption of normality is not satisfied for many complex multi-variate datasets which are generated in various fields. In such situation, traditionalparametric inferential procedures are not appropriate. Pesarin(2001) has proposedNPC (nonparametric combination) theory for a dependent tests. This theory iswidely used in multi aspect testing. NPC divides a global null hypothesis into sev-eral partial null hypotheses and then use a Fisher’s permutation principle to findthe p-value of each of these partial null hypotheses. A suitable combining functionis used to combine the p-values of partial null hypotheses. The null distribution ofthe choosen combining function is obtained by using the Fisher permutation prin-ciple to compute the global p-value. Based on this global p-value, decide whetherto accept or reject the global null hypothesis. In the literature, less number of non-parametric multivariate two sample tests are proposed for simultaneously testinglocation and scale parameters of two multivariate distributions. Also, two sampletests based on the combination of nonparametric multivariate location test andscale test are appeared less in the literature. In the present article, we have usednonparametric combination (NPC) theory to develop nonparametric multivariatetwo-sample tests based on data depth for simultaneously testing location and scaleparameters of two multivariate distributions. Data-depth based two-sample testsfor location and scale are combined using an appropriate combining function toproduce a new test for testing both location and scale parameters. Fisher’s per-mutation test is used to compute the p-value of the tests. Monte Carlo simulationsare used to obtain the empirical power of the proposed tests and performance ofproposed tests is compared with the H-test provided by Chenouri & Small (2012).The rest of the article is organized as follows.

A review of combining functions is given in Section 2. In Section 3, we discussthe notion of data depth and some well-known data depth functions. Tests fortesting differences between locations and scales based on the notion of data depthare discussed in Section 4. We describe the proposed tests in Section 5. Simulation



results are reported in Section 6. Illustration with a real-life data is provided inSection 7 and concluding remarks are given in the last Section.

2. Combining Functions

In a meta-analysis, combining functions plays an important role while summa-rizing all the results obtained from various independent groups. Several combiningfunctions are available in the literature, some of which are discussed in the follow-ing subsections.

2.1. Fisher’s Combining Function

Let pi, i = 1, 2, . . . , k denote the p-value of ith hypothesis test. Then theFisher’s combining function (Fisher 1925) is denoted by Fc and is defined as,

Fc = −2∑k

i=1 loge pi.

If all the null hypotheses are true and pi are independent then Fc follows a chi-square distribution with 2k degrees of freedom, where k is the number of testsbeing combined. Under the null hypothesis of each individual test, p-value followscontinuous uniform distribution over interval [0, 1]. Therefore, the test statisticFc follows chi-square distribution with 2k degrees of freedom. If all the p-valuesare small, then the quantity Fc is large. Smaller p-values indicate the rejectionof individual test. As a result of this, test statistic Fc rejects the global nullhypothesis.

2.2. Tippet Combining Function

The Tippet combining function (Tippett 1952) is denoted by Tc and is definedas,

Tc = max1≤i≤k

(1− pi).

If all the null hypotheses are true and pi are independent, distribution of Tc be-haves according to the largest (smallest) of k random values from the uniformdistribution over (0, 1). As smaller the p-value (pi) of ith individual test, thequantity (1− pi) is large. Therefore, the test rejects the global null hypothesis forthe larger value of Tc.

2.3. Liptak Combining Function

The Liptak combining function (Liptak 1958) is denoted by Lc and is definedas,

Lc =∑k

i=1 Φ−1(1− pi),



where Φ(.) is the distribution function of standard normal distribution. As smallerthe p-value (pi) of ith individual test, the quantity Φ−1(1− pi) is large. Thereforethe test rejects the global null hypothesis for the larger value of Lc. If all the nullhypotheses are true and pi are independent then Lc follows the normal distributionwith mean 0 and variance k.

In the next section, we discuss the notion of data depth.

3. Notion of Data Depth

Let (X1, X2, . . . , Xm) be a data set (cloud), where each Xi ∈ Rp, i = 1, 2, . . . ,mis assumed to follow a continuous distribution with cumulative distribution func-tion (CDF) F (.). Let D(x, F ) be the depth of a point x with respect to F (.). Adata depth is a non-negative function defined from Rp to [0,∞). The notion ofdata depth can be used to obtain the location of a given data points with respectto a data cloud. It measures the centrality of a given data point with respect to agiven distribution F (.) or data cloud. Data depth gives a natural center-outwardranking to a data points with respect to data cloud. Such rankings were used fortesting difference in the location and scale parameters of two or more multivariatedistributions, constructing nonparametric control charts, outliers detection, andclassification problems etc.

In the following, we describe three well-known depth functions. However, weuse simplicial and Tukey’s halfspace depth functions for our discussion throughout this article.

• Simplicial Depth

The simplicial depth (Liu 1990) for any point x ∈ Rp with respect to F (.) on Rpis denoted by SD(x, F ) and is given by,

SD(x, F ) = PF (s[X1, X2, . . . , Xp+1] ∋ x),

where, X1, X2, . . . , Xp+1 are independent and identically distributed observationsfrom F and s[X1, X2, . . . , Xp+1] is a closed simplex formed by X1, X2, . . . , Xp+1.The sample version of simplicial depth can be obtained by replacing F by Fm inthis expression. That is,

SD(x, Fm) =(

mp+1

)−1 ∑∗ I(xϵs[Xi1, Xi2, . . . , Xip+1]),

where (∗) runs over all possible subsets of X1, X2, . . . , Xm of size (p+ 1) and I(.)is an usual indicator function.

• Tukey’s Halfspace Depth

The Tukey’s halfspace depth (Tukey 1975) for any point x ∈ Rp with respect toF (.) on Rp is denoted by HSD(x, F ) and is given by,



HSD(x, F ) = infH{P (H) : H is a closed halfspace containing x},

where P (.) is a probability. The sample version of HSD(x, F ) is obtained byreplacing F by Fm. It is a smallest fraction of the data points contained in aclosed halfspace which contains x. That is,

HSD(x, Fm) =min||l||=1#{i : l′xi ≥ l′x}

m.

If p = 1 then HSD(x, F ) = min{F (x), 1− F (x−)}.

• Mahalanobis Depth

The Mahalanobis depth (Mahalanobis 1936) for any point x ∈ Rp with respect toF (.) on Rp is denoted by MD(x, F ) and is given by,

MD(x, F ) = 11+(x−µ)′Σ−1(x−µ) ,

where µ and Σ are the location parameter (center) and the variance-covariancematrix (dispersion matrix) of F (.). The quantity (x − µ)′Σ−1(x − µ) is a Maha-lanobis distance of a point x from µ. That is Mahalanobis depth is defined by usingMahalanobis distance. The sample version of Mahalanobis depth can be obtainedby replacing µ and Σ by X̄ (sample mean) and S (sample variance-covariancematrix).

In the next section, we discuss tests for testing differences between locationparameters and scale parameters based on the notion of data depth.

4. Tests Based on Data Depth

Let (X1, X2, . . . , Xm) and (Y1, Y2, . . . , Yn) be two independent random samplesfrom two continuous distributions F (.) and G(.) respectively, where Xi, Yj ∈ IRp,i = 1, 2, . . . ,m and j = 1, 2, . . . , n. Let D(x, F ) and D(x,G) be the depths of apoint x ∈ Z with respect to F (.) and G(.) respectively, where Z = X ∪ Y . A setcontaining such points is defined as,

DD(F,G) = {(D(x, F ), D(x,G)), ∀ x ∈ Z}.

The empirical version of DD(F,G) based on the above described two randomsamples is given by,

DD(Fm, Gn) = {(D(x, Fm), D(x,Gn)), ∀ x ∈ Z}.

DD plot is a scatter plot, which is the plot of points in the set DD(Fm, Gn).The DD plot can be used for comparing two multivariate samples by graphi-

cally. The difference between locations or scales or skewness or kurtosis is associ-ated with different pictures observed on the DD plots. If F and G are identical



then the points should fall on a 450 line segment on the empirical DD Plot. This isillustrated in Figure 1(a), which is the DD plot of two multivariate samples drawnfrom the bivariate normal distribution with mean vector µ = 0

¯and variance-

covariance matrix I2, where I2 is the identity matrix of order two. The divergenceof F from G will indicate divergence of points from 450 line segment and Figure1(b), Figure 2(a), Figure 2(b) and Figure 3 reveal different pictures of DD plotthat indicate the location differences, large location differences, scale differencesand skewness differences (both location and scale differences) respectively. TheDD plot in Figure 1(b) has a leaf-shaped picture with the cusp lying on the di-agonal line towards the upper right corner and the leaf steam at the lower leftcorner point (0, 0), when there is a shift in location parameters of two multivariatesamples. In each of these Figures, we plot DD plot of DG against DF where Fand G have chosen appropriately, where DF and DG are the depth of the pointswith respect to F and G respectively. We use simplicial depth as a depth functionto plot the DD plot in Figure 1-3. The DD plots have been plotted using ’depth’package available in R.

Figure 1: DD plots of (a) identical distributions and (b) location shift.

In the following subsections, we review four tests for testing equality of loca-tions and a test for testing scale differences between two multivariate distributions.

4.1. Nonparametric Tests for Location Differences BetweenTwo Multivariate Distributions

Li & Liu (2004) proposed two tests viz. T−based and M−based tests for testinglocation differences between two multivariate distributions based on DD-plot.

• T−based test

In the presence of location shift in two distribution, the DD plot has a leaf-shaped picture (Figure 1(b), Figure 2(a)) with the leaf stem anchoring at the lower



Figure 2: DD plots of (a) large location shift and (b) scale increase.

Figure 3: DD plot of skewness difference.

left corner point (0, 0) and the cusp lying on the diagonal line pointing towards theupper right corner. On the basis of this observation, Li & Liu (2004) constructedthe test statistic which is the distance between the origin (0, 0) and the cusppoint. Li & Liu (2004) suggested the procedure to calculate the distance betweenthe cusp point and the origin (0, 0). Smaller the distance indicates the larger shiftin location. The p-value of the test is obtained by using the Fisher’s permutationtest.

• M−based test

In the literature of data depth, the point having largest depth is called as thelocation parameter or the deepest point. Therefore if the two distributions F andG are identical then they should have the same deepest point. In case of locationshift, the deepest point with respect to the distribution F would not be the deepestpoint with respect to the distribution G. In fact, the deepest point of F will have



a smaller depth value with respect to G. M -based test statistic due to Li & Liu(2004) is given by,

M = min{D(v, Fm), D(u,Gn)},

where v is the deepest point of X ∪ Y corresponding to Gn, and u is the deepestpoint of X ∪ Y corresponding to Fm. Here smaller the value of M , stronger theevidence against H0. The p-value of the test is determined by using the Fisher’spermutation test.

Li et al. (2011) have proposed two depth based Kolmogorov-Smirnov (KS) andCramer-Von Mises (CM) type tests for comparing species assemblages. The teststatistics are defined as follows,

• KS type test statistic

TKS =∑m+n

i=1 |D(zi, Fm)−D(zi, Gn)|

• CM type test statistic

TCM =∑m+n

i=1 (D(zi, Fm)−D(zi, Gn))2,

where, zi is the ith point in X ∪ Y . These tests reject H0 for large values of TKSand TCM . Larger the value of TKS and TCM , stronger is the evidence against H0and it represents larger location shift between the two distributions. The p-valueof these tests are also obtained by using the Fisher’s permutation test.

4.2. Nonparametric Test for Testing Equality of ScaleParameters of Two Multivariate Distributions

Li & Liu (2016) proposed a test for testing scale differences between two mul-tivariate distributions and the test statistic is defined as,

S =∑

z∈X∪Y(D(z, Fm)−D(z,Gn)),

where z is the point in X ∪Y . Larger the value of S, stronger the evidence againstH0. The p-value of this test is also obtained by using Fisher’s permutation test.

In the following section, we propose tests for simultaneously testing locationand scale parameters of two multivariate distributions using NPC theory.

5. Proposed Tests

Let (X1, X2, . . . , Xm) and (Y1, Y2, . . . , Yn) be the two independent random sam-ples of size m and n from two continuous distribution with cumulative distribu-tion functions F (.) and G(.) respectively, where each Xi, Yj ∈ Rp, i = 1, 2, . . . ,m,j = 1, 2, . . . , n. We wish to test the null hypothesis,



H0 : F (x) = G(x) ∀ x = (x1, x2, . . . , xp)T ,

against an alternative hypothesis

H1 : F (x) = G(x−µσ ) ∀ x = (x1, x2, . . . , xp)

T ,

where, (x−µσ ) =(

(x1−µ1)σ1

, (x2−µ2)σ2 , . . . ,(xp−µp)

σp

)T, µ = (µ1, µ2, . . . , µp)T , σ =

(σ1, σ2, . . . , σp)T and σ is a diagonal matrix.

It is equivalent to test

H0 : {µ = 0} ∩ {σ = 1}, (1)

against

H1 : {µ ̸= 0} ∪ {σ ̸= 1}.

We need a simultaneous test procedure for testing both location and scale pa-rameters to test null hypothesis in (1). Pesarin (2001) has developed a theoryof NPC of dependent tests. Such theory is used to develop a nonparametric testfor two-sample locations and/or scales problem. NPC assumes a global null hy-pothesis of both location and scale parameters of two multivariate distributionsare same. This global null hypothesis is divided into two partial null hypotheses,one for location problem and other for scale problem. NPC then uses a Fisher’spermutation test to obtain the single p-value of global null hypothesis and par-tial p-values for each partial null-hypothesis. Based on NPC theory, Park (2015)provided several nonparametric simultaneous test procedures for the univariatetwo-sample location-scale problem using combining functions.

We use the same NPC theory for developing tests for simultaneously testinghypothesis given in (1). We use separate test statistic for testing the sub-nullhypotheses H01 : µ = 0 and H02 : σ = 1 and then combine the p-values of thesetwo separate tests to obtain the global p-value of the test. Suppose the statisticTL is used for location test and TS is used for scale test and pL and pS be thep-values of the location test and scale test respectively. Then for obtaining theglobal p-value for simultaneously testing (1), we use an appropriate combiningfunction to combine the p-values pL and pS .

The null distribution of the selected combined function is obtained by usingFisher’s permutation principle to compute the global p-value of the location-scaletest. An algorithm for obtaining global p-value is given below.

Algorithm for Obtaining Global p-value

The following algorithm is required to get the global p-value of the location-scale test by using NPC theory.



Algorithm1: Divide the global null hypothesis that is location-scale problem in the partial

null hypotheses. That is, H01 : µ = 0 and H02 : σ = 1.2: Select an appropriate test statistic for each partial null hypotheses H01 and

H02. That is TL and TS , which are sensitive to the alternative hypothesis.3: Calculate its value for observed data, which is denoted by (TLb=0, TSb=0).4: Take B permutations of the original samples and calculate the value of each

test statistic for permuted data, (TLb , TSb ), b ∈ {1, 2, . . . , B}.5: Compute the permutation p-values for partial null hypotheses that is partial

p-values, (pLb=0, pSb=0). If we reject null hypothesis for larger (or smaller) valuesof test statistic then we use following formula

pLb=0 =[1+

∑I(TLb ̸=0≥T

Lb=0)]

(B+1)(or pLb=0 =

[1 +∑

I(TLb̸=0 ≤ TLb=0)](B + 1)

). (2)

Similarly, we calculate pSb=0.6: Compute partial pseudo p-value (pLb , pSb ) for each permutation. If we reject

null hypothesis for larger (or smaller) values of test statistic then we use fol-lowing formula

pLk =[1+

∑I(TLb ̸=k≥T

Lb=k)]

(B+1) , k ∈ {1, 2, . . . , B}(or pLk =

[1 +∑

I(TLb̸=k ≤ TLb=k)](B + 1)

, k ∈ {1, 2, . . . , B}). (3)

Similarly, we define pSk for scale test.7: Use an appropriate combining function to combine p-values to get a single

global test statistic. We get global test statistic as,• For Fisher’s combining function

T globalb = −2(logepLb + logep

Sb ). (4)

• For Tippet combining function

T globalb = max{(1− pLb ), (1− pSb )}. (5)

• For Liptak combining function

T globalb = Φ−1(1− pLb ) + Φ−1(1− pSb ). (6)

8: By combining the p-values, a final vector of length (B + 1) is produced. Thefirst element T globalb=0 summarizes the partial p-values (pLb=0, pSb=0) observed onthe initial data sets and the remaining elements T globalb , b ∈ {1, 2, . . . , B} areproduced by combining the corresponding pseudo p-values.

9: Similarly to the partial p-values, calculate a p-value of the global test by usingthe following equation



pglobal =[1 +

∑I(T globalb̸=0 ≥ T

globalb=0 )]

(B + 1)

10: Take decision about acceptance or rejection of the global null hypothesis (1)by using global p-value.

In the next section, we evaluate the performance of the proposed tests anddata depth based H-test proposed by Chenouri & Small (2012) in terms of theempirical power.

6. Simulation Study

The performance of the proposed tests is investigated through Monte Carlosimulation experiment in terms of empirical power for different bivariate symmetric(normal and t) and bivariate skewed (normal and t) distributions, which are listedin Table 1. The parameter µ represents the shift in the location parameter, σrepresents the change in scale parameter, a represents the shape parameter (orskewness parameter) and v represents the degrees of freedom. The number ofobservations generated from each of F and G are taken to be m = n = 50 andm = n = 100 respectively. The p-value of the test is obtained by permutingoriginal samples 500 times and the power is obtained by the proportion of timesthe p-values are less than or equal to the nominal level of significance α. Anempirical power of the proposed tests is compared to the H-test. All the resultsin the Table 3 to Table 10 are reported for 1000 Monte Carlo simulations andR-software is used for simulation studies.

We use KS-type, CM-type (Li et al. 2011), T-based and M-based tests (Li &Liu 2004) for a location problem and S-test (Li & Liu 2016) for a scale problem.Simplicial and Tukey’s halfspace depth functions are used in our study to computethe depth of an observation. The Fisher and Tippet combining functions are usedto combine the p-values of location test and scale test. The results are reportedfor all combinations of various values of µ = b ∗ (1, 1) and σ = c ∗ (1, 1) whichare given in Table 2. A distribution F always corresponds to value of µ = (0, 0)and σ = (1, 1) and the distribution G corresponds to any value of µ and σ in theTable 2.

Table 1: Distributions used in the simulation study.

Distribution Under H0 Under H1Symmetric normal N2(µ = (0, 0), σ = (1, 1)) N2(µ, σ)Symmetric t t2(µ = (0, 0), σ = (1, 1)) t2(µ, σ)Skewed normal SN2(µ = (0, 0), σ = (1, 1), a = (10, 4)T ) SN2(µ, σ, a = (10, 4)T )Skewed t ST2(µ = (0, 0), σ = (1, 1), a = (10, 4)T , v = 3) ST2(µ, σ, a = (10, 4)T , v = 3)



Table 2: Values of b and c.b 0.0 0.2 0.4 0.6 0.8 1.0

c 1.0 1.2 1.4 1.6 1.8 2.0

Simulation results with sample sizes m = n = 50 and m = n = 100 forsimplicial depth are reported in Table 3 to Table 6 and for Tukey’s halfspacedepth, results are reported in Table 7 to Table 10. In the Table 3 to Table 10,T1, T2, T3 and T4 indicates proposed test using the combination of KS and Stests, CM and S tests, T-based and S tests and M-based and S tests respectively.In the following, we have given conclusions which are obtained for a sample sizem = n = 100. The conclusions reached with the sample sizes m = n = 100 aresimilar to those reached in the m = n = 50. The size of the proposed tests isobtained and it is shown in the first row of Table of each simulation scenario.

6.1. Simulation Results for Symmetric Normal Distribution

In this case, the proposed tests have greater empirical power as compared tothe H-test for both depth functions on a shift by shift increase in the locationparameter as well as in the scale parameter. The tests T1 and T2 perform betterthan the T3 and T4 tests for both Fisher’s and Tippet combining functions. Notethat, the proposed tests with Fisher’s combining function work better than thatof the Tippet combining function. The performance of the proposed tests withsimplicial depth is similar to the halfspace depth.

6.2. Simulation Results for Symmetric t Distribution

Also, in this case, the proposed tests perform better than the H-test. Thetests T2, T3 and T4 are more powerful than T1 test for both Fisher’s and Tippetcombining functions with simplicial depth. But, for halfspace depth, all the T1,T2, T3 and T4 tests perform equally well. Also, the proposed tests with Fisher’scombining function have greater power than that of the Liptak combining functionfor both simplicial depth and halfspace deph.

6.3. Simulation Results for Skewed Normal and Skewed tDistributions

In this case, the proposed tests with Fisher’s and Tippet combining functionshave greater empirical power as compared to the H-test for a small shift in thelocation parameteras well as small change in the scale parameter. For a large shift,proposed tests and H-test perform equally well for both combining functions withsimplcial and halfspace depths. The tests T1 and T2 are more powerful than theT3 and T4 tests for a small shift in location parameter and a small change in scaleparameter with simplcial and halfspace depths.

The performance of the proposed tests for skewed t distribution is same as inthe case of skewed normal distribution.



Tab

le3:

Sim

ulat

edpo

wer

sof

prop

osed

test

sfo

rno

rmal

dist

ribu

tion

wit

hsa

mpl

esi

zesm

=n=

50

andm

=n=

100.

cb

m=

n=

50

m=

n=

100

Fis

her

Tip

pet

H-t

est

Fis

her

Tip

pet

H-t

est

T1

T2

T3

T4

T1

T2

T3

T4

T1

T2

T3

T4

T1

T2

T3

T4

1.0

0.0

0.05

20.

055

0.05

30.

050

0.04

90.

053

0.05

00.

053

0.04

80.

048

0.04

50.

053

0.04

80.

049

0.04

70.

048

0.04

80.

051

0.2

0.11

50.

148

0.10

80.

116

0.12

90.

153

0.10

30.

127

0.14

00.

190

0.26

10.

188

0.20

90.

217

0.29

30.

183

0.20

70.

168

0.4

0.37

00.

514

0.34

80.

362

0.41

90.

557

0.34

20.

373

0.43

90.

721

0.86

70.

703

0.71

50.

784

0.89

30.

719

0.74

30.

708

0.6

0.76

50.

912

0.73

40.

749

0.82

70.

933

0.75

80.

783

0.82

90.

988

0.99

90.

985

0.98

60.

997

1.00

00.

989

0.98

90.

992

0.8

0.97

00.

995

0.94

80.

956

0.98

40.

998

0.95

90.

971

0.97

71.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.0

1.00

01.

000

0.99

90.

999

1.00

01.

000

0.99

90.

999

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.2

0.0

0.16

10.

149

0.14

20.

146

0.15

00.

146

0.14

30.

148

0.06

80.

272

0.25

90.

240

0.24

10.

240

0.23

80.

235

0.23

80.

162

0.2

0.23

80.

274

0.21

60.

222

0.20

30.

238

0.19

10.

193

0.15

60.

419

0.47

90.

382

0.40

50.

360

0.41

60.

326

0.33

90.

284

0.4

0.49

50.

608

0.45

10.

480

0.46

90.

579

0.39

60.

422

0.44

10.

837

0.91

90.

792

0.82

30.

824

0.90

30.

730

0.76

40.

741

0.6

0.79

30.

909

0.76

60.

789

0.80

90.

917

0.74

50.

766

0.79

80.

988

0.99

80.

984

0.98

00.

990

0.99

70.

980

0.97

30.

979

0.8

0.96

50.

993

0.95

10.

956

0.97

00.

995

0.94

30.

950

0.96

81.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.0

0.99

61.

000

0.99

80.

997

0.99

71.

000

0.99

70.

996

0.99

71.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.4

0.0

0.35

40.

336

0.32

30.

322

0.33

40.

326

0.32

00.

324

0.13

50.

611

0.60

90.

553

0.55

40.

591

0.58

40.

580

0.58

00.

412

0.2

0.42

20.

439

0.38

40.

389

0.36

80.

381

0.34

50.

346

0.20

60.

712

0.74

40.

651

0.67

80.

636

0.65

30.

592

0.61

10.

522

0.4

0.61

90.

706

0.57

40.

597

0.54

20.

614

0.46

90.

490

0.45

80.

929

0.95

70.

880

0.89

50.

885

0.93

20.

810

0.82

10.

813

0.6

0.85

40.

931

0.81

20.

828

0.82

80.

908

0.73

40.

757

0.78

10.

995

1.00

00.

992

0.99

10.

995

1.00

00.

985

0.98

00.

986

0.8

0.97

70.

999

0.96

10.

963

0.97

50.

996

0.94

10.

946

0.96

01.

000

1.00

01.

000

1.00

01.

000

1.00

00.

999

1.00

01.

000

1.0

0.99

81.

000

0.99

80.

998

0.99

61.

000

0.99

80.

994

0.99

81.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.6

0.0

0.55

40.

551

0.51

80.

522

0.54

90.

547

0.54

30.

541

0.22

00.

854

0.85

30.

827

0.82

40.

857

0.85

20.

849

0.85

00.

690

0.2

0.61

80.

630

0.58

00.

588

0.56

80.

569

0.54

80.

549

0.28

50.

892

0.90

30.

869

0.85

80.

857

0.86

20.

839

0.84

00.

745

0.4

0.75

40.

803

0.70

40.

712

0.66

80.

718

0.60

30.

620

0.50

40.

977

0.98

60.

949

0.96

40.

957

0.97

00.

914

0.92

80.

900

0.6

0.92

60.

968

0.88

50.

900

0.88

40.

937

0.79

90.

833

0.82

10.

998

1.00

00.

996

0.99

70.

995

0.99

90.

983

0.98

50.

987

0.8

0.97

70.

995

0.97

10.

969

0.97

00.

990

0.94

00.

943

0.95

21.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.0

0.99

81.

000

0.99

70.

999

0.99

81.

000

0.99

20.

996

0.99

91.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.8

0.0

0.73

00.

714

0.69

30.

695

0.73

00.

726

0.72

40.

725

0.35

10.

957

0.95

90.

949

0.94

50.

963

0.96

00.

960

0.96

10.

852

0.2

0.76

60.

778

0.73

10.

739

0.73

80.

742

0.72

00.

733

0.39

70.

972

0.97

50.

957

0.96

50.

962

0.96

30.

956

0.95

90.

894

0.4

0.86

50.

895

0.82

40.

837

0.79

70.

830

0.74

60.

762

0.59

50.

994

0.99

70.

989

0.99

00.

983

0.98

90.

969

0.97

40.

955

0.6

0.94

60.

974

0.91

60.

939

0.90

70.

937

0.85

20.

873

0.82

30.

999

1.00

00.

999

0.99

90.

998

1.00

00.

992

0.99

60.

993

0.8

0.98

90.

998

0.97

80.

982

0.98

10.

993

0.94

80.

952

0.95

21.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.0

1.00

01.

000

0.99

50.

998

0.99

91.

000

0.98

90.

990

0.99

21.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

2.0

0.0

0.86

80.

863

0.83

70.

836

0.86

60.

863

0.86

20.

862

0.47

30.

990

0.99

10.

984

0.98

40.

992

0.99

20.

992

0.99

20.

938

0.2

0.87

60.

875

0.84

90.

844

0.84

30.

846

0.83

10.

834

0.51

70.

997

0.99

80.

998

0.99

40.

995

0.99

50.

994

0.99

40.

959

0.4

0.92

40.

945

0.88

80.

911

0.88

10.

895

0.84

70.

863

0.65

40.

998

1.00

00.

996

0.99

70.

996

0.99

60.

992

0.99

40.

984

0.6

0.97

20.

986

0.94

90.

958

0.94

10.

961

0.89

40.

907

0.81

81.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

0.99

90.

995

0.8

0.99

40.

999

0.98

50.

988

0.98

40.

993

0.96

20.

962

0.96

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.0

1.00

01.

000

0.99

60.

998

0.99

91.

000

0.99

00.

990

0.99

31.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000



Tab

le4:

Sim

ulat

edpo

wer

sof

prop

osed

test

sfo

rt

dist

ribu

tion

wit

hsa

mpl

esi

zesm

=n=

50

andm

=n=

100.

cb

m=

n=

50

m=

n=

100

Fis

her

Tip

pet

H-t

est

Fis

her

Tip

pet

H-t

est

T1

T2

T3

T4

T1

T2

T3

T4

T1

T2

T3

T4

T1

T2

T3

T4

1.0

0.0

0.06

00.

055

0.04

70.

048

0.05

10.

046

0.04

50.

053

0.04

90.

048

0.05

00.

051

0.05

60.

049

0.05

20.

054

0.05

30.

050

0.2

0.07

60.

099

0.09

90.

096

0.08

50.

107

0.10

30.

101

0.11

40.

133

0.18

30.

167

0.18

10.

152

0.19

40.

172

0.17

40.

150

0.4

0.21

90.

323

0.28

00.

292

0.24

80.

355

0.29

40.

308

0.41

50.

448

0.61

00.

623

0.60

20.

526

0.67

20.

633

0.62

90.

567

0.6

0.50

70.

692

0.65

40.

627

0.60

50.

754

0.67

80.

664

0.77

90.

874

0.95

60.

955

0.94

90.

920

0.97

30.

961

0.96

20.

952

0.8

0.79

30.

927

0.90

10.

870

0.85

20.

947

0.91

90.

903

0.95

90.

989

1.00

00.

999

0.99

90.

993

1.00

00.

999

0.99

90.

998

1.0

0.94

30.

989

0.98

40.

975

0.96

90.

995

0.98

60.

984

0.99

61.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.2

0.0

0.11

60.

118

0.11

40.

109

0.10

30.

102

0.10

30.

104

0.07

60.

186

0.18

50.

156

0.16

10.

160

0.15

50.

154

0.15

40.

123

0.2

0.16

40.

168

0.16

00.

163

0.13

80.

148

0.14

10.

146

0.13

20.

255

0.29

10.

275

0.29

40.

226

0.25

90.

235

0.24

60.

196

0.4

0.28

20.

381

0.35

70.

355

0.26

10.

362

0.30

70.

304

0.38

00.

563

0.69

10.

671

0.68

70.

547

0.67

00.

624

0.63

30.

566

0.6

0.53

30.

702

0.65

80.

646

0.55

00.

707

0.62

70.

615

0.75

50.

886

0.95

90.

955

0.94

90.

906

0.96

30.

945

0.94

10.

930

0.8

0.78

10.

916

0.88

90.

855

0.81

60.

923

0.87

80.

864

0.94

40.

990

0.99

81.

000

0.99

60.

995

0.99

91.

000

0.99

70.

999

1.0

0.93

20.

986

0.97

40.

967

0.94

40.

990

0.97

90.

973

0.99

21.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.4

0.0

0.22

50.

219

0.19

60.

216

0.20

50.

198

0.20

00.

202

0.11

20.

386

0.37

80.

340

0.34

00.

359

0.35

30.

349

0.34

80.

304

0.2

0.30

10.

316

0.27

60.

291

0.26

70.

277

0.25

70.

267

0.19

10.

483

0.51

40.

474

0.48

90.

425

0.44

00.

422

0.42

60.

398

0.4

0.40

60.

491

0.44

40.

455

0.34

50.

417

0.35

80.

379

0.41

60.

722

0.80

30.

779

0.80

40.

671

0.74

80.

691

0.71

30.

697

0.6

0.60

40.

740

0.69

60.

694

0.57

10.

699

0.62

80.

620

0.72

40.

926

0.96

70.

962

0.96

50.

921

0.96

20.

946

0.94

20.

946

0.8

0.81

70.

927

0.89

50.

890

0.81

90.

919

0.86

90.

862

0.94

00.

991

0.99

80.

997

0.99

80.

992

0.99

90.

997

0.99

70.

996

1.0

0.93

40.

987

0.97

80.

971

0.94

40.

988

0.97

10.

964

0.99

00.

999

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.6

0.0

0.38

90.

381

0.33

40.

351

0.35

60.

349

0.34

20.

350

0.20

90.

614

0.60

70.

556

0.56

40.

591

0.58

60.

582

0.58

40.

507

0.2

0.41

10.

419

0.37

90.

394

0.36

30.

365

0.33

70.

353

0.25

90.

661

0.67

60.

641

0.65

40.

619

0.61

90.

599

0.61

00.

565

0.4

0.54

00.

603

0.54

80.

567

0.45

90.

516

0.46

90.

482

0.47

30.

829

0.87

90.

852

0.86

50.

771

0.82

50.

776

0.78

30.

790

0.6

0.70

10.

798

0.75

00.

754

0.63

50.

746

0.66

30.

660

0.75

40.

950

0.98

00.

979

0.97

60.

933

0.96

50.

948

0.94

80.

956

0.8

0.84

10.

928

0.90

00.

886

0.82

30.

913

0.86

20.

853

0.92

60.

994

0.99

90.

998

0.99

90.

992

0.99

70.

993

0.99

70.

996

1.0

0.94

40.

987

0.98

30.

972

0.94

10.

986

0.97

40.

963

0.98

81.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.8

0.0

0.49

10.

486

0.44

70.

454

0.47

00.

464

0.45

90.

462

0.28

90.

780

0.77

80.

725

0.72

80.

770

0.76

30.

754

0.75

90.

682

0.2

0.56

40.

575

0.53

10.

541

0.52

30.

527

0.50

50.

514

0.36

20.

841

0.84

90.

796

0.80

50.

798

0.79

70.

783

0.78

80.

759

0.4

0.62

70.

668

0.64

60.

636

0.54

40.

582

0.55

30.

551

0.51

20.

903

0.92

40.

900

0.91

50.

844

0.86

80.

851

0.85

30.

870

0.6

0.76

20.

839

0.79

40.

804

0.67

90.

773

0.69

60.

711

0.77

70.

976

0.99

40.

990

0.99

00.

959

0.98

00.

970

0.97

10.

979

0.8

0.86

80.

950

0.92

80.

922

0.83

60.

924

0.87

40.

866

0.93

00.

997

1.00

00.

999

0.99

90.

995

1.00

00.

997

0.99

70.

998

1.0

0.95

30.

986

0.98

10.

976

0.94

10.

982

0.96

40.

959

0.98

71.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

2.0

0.0

0.64

80.

626

0.59

20.

608

0.62

10.

611

0.60

80.

615

0.39

30.

905

0.90

50.

879

0.87

50.

900

0.89

70.

891

0.89

00.

853

0.2

0.66

30.

669

0.63

00.

640

0.62

80.

621

0.61

90.

617

0.43

30.

914

0.92

30.

889

0.89

90.

901

0.90

00.

886

0.88

80.

867

0.4

0.72

10.

751

0.70

90.

732

0.63

90.

657

0.62

80.

645

0.58

00.

959

0.97

30.

963

0.96

40.

931

0.94

00.

925

0.93

00.

937

0.6

0.81

80.

871

0.83

00.

849

0.74

60.

804

0.73

90.

752

0.78

40.

990

0.99

40.

994

0.99

40.

978

0.98

80.

974

0.97

80.

991

0.8

0.91

20.

964

0.94

00.

943

0.87

50.

929

0.88

50.

891

0.93

81.

000

1.00

01.

000

0.99

90.

998

1.00

00.

998

0.99

70.

999

1.0

0.96

60.

989

0.98

40.

985

0.95

70.

984

0.96

50.

970

0.98

81.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000



Tab

le5:

Sim

ulat

edpo

wer

sof

prop

osed

test

sfo

rsk

ewed

norm

aldi

stri

buti

onw

ith

sam

ple

size

sm

=n=

50

andm

=n=

100.

cb

m=

n=

50

m=

n=

100

Fis

her

Tip

pet

H-t

est

Fis

her

Tip

pet

H-t

est

T1

T2

T3

T4

T1

T2

T3

T4

T1

T2

T3

T4

T1

T2

T3

T4

1.0

0.0

0.05

20.

056

0.05

00.

053

0.04

70.

048

0.05

00.

053

0.04

30.

051

0.04

80.

054

0.05

30.

054

0.05

30.

057

0.05

70.

047

0.2

0.28

60.

328

0.13

50.

184

0.41

70.

450

0.18

70.

254

0.28

20.

728

0.76

20.

335

0.40

70.

855

0.87

70.

448

0.52

20.

605

0.4

0.96

10.

960

0.69

30.

790

0.98

40.

984

0.78

50.

858

0.86

61.

000

1.00

00.

979

0.98

51.

000

1.00

00.

990

0.99

30.

999

0.6

1.00

01.

000

0.99

20.

996

1.00

01.

000

0.99

70.

998

0.99

81.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

0.8

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.0

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.2

0.0

0.14

20.

144

0.13

20.

129

0.12

10.

123

0.12

00.

122

0.07

70.

214

0.22

10.

192

0.20

30.

180

0.18

90.

175

0.17

60.

134

0.2

0.40

90.

536

0.27

90.

316

0.52

90.

621

0.31

80.

375

0.34

60.

841

0.89

50.

521

0.57

20.

911

0.94

20.

599

0.65

60.

647

0.4

0.98

00.

984

0.79

80.

872

0.99

10.

994

0.85

20.

917

0.90

21.

000

1.00

00.

990

0.99

41.

000

1.00

00.

997

0.99

81.

000

0.6

1.00

01.

000

0.99

10.

998

1.00

01.

000

0.99

51.

000

0.99

91.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

0.8

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.0

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.4

0.0

0.28

80.

310

0.25

70.

275

0.24

40.

261

0.23

50.

253

0.12

90.

478

0.53

30.

439

0.44

50.

425

0.44

60.

401

0.39

40.

317

0.2

0.58

20.

704

0.40

00.

480

0.62

50.

741

0.41

00.

478

0.46

60.

937

0.97

50.

748

0.79

10.

950

0.98

10.

763

0.80

80.

771

0.4

0.98

90.

991

0.85

10.

911

0.99

50.

997

0.89

30.

937

0.92

51.

000

1.00

00.

996

0.99

91.

000

1.00

00.

998

0.99

91.

000

0.6

1.00

01.

000

0.99

50.

998

1.00

01.

000

0.99

50.

998

0.99

91.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

0.8

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.0

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.6

0.0

0.45

20.

495

0.40

80.

430

0.39

70.

423

0.37

30.

389

0.22

60.

741

0.79

30.

692

0.70

90.

672

0.71

20.

637

0.65

20.

546

0.2

0.72

30.

852

0.54

20.

593

0.73

40.

847

0.51

10.

558

0.56

20.

984

0.99

60.

894

0.90

80.

982

0.99

70.

876

0.88

60.

894

0.4

0.99

10.

998

0.91

30.

938

0.99

60.

999

0.93

20.

952

0.95

21.

000

1.00

01.

000

0.99

91.

000

1.00

01.

000

1.00

01.

000

0.6

1.00

01.

000

0.99

80.

999

1.00

01.

000

0.99

91.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

0.8

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.0

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.8

0.0

0.61

00.

654

0.54

30.

577

0.52

50.

567

0.48

00.

500

0.31

90.

909

0.93

80.

875

0.88

30.

858

0.90

50.

828

0.83

80.

755

0.2

0.79

60.

905

0.67

60.

715

0.78

80.

909

0.61

50.

660

0.65

80.

992

0.99

80.

950

0.96

50.

991

0.99

70.

928

0.94

40.

947

0.4

0.99

81.

000

0.95

40.

961

0.99

91.

000

0.95

80.

963

0.96

91.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

0.6

1.00

01.

000

0.99

90.

999

1.00

01.

000

0.99

90.

999

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

0.8

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.0

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

2.0

0.0

0.77

30.

812

0.71

90.

738

0.69

00.

739

0.65

20.

660

0.46

20.

977

0.99

00.

958

0.96

50.

953

0.97

40.

936

0.94

50.

885

0.2

0.89

40.

961

0.79

30.

819

0.86

50.

953

0.70

00.

744

0.72

61.

000

1.00

00.

987

0.99

00.

998

1.00

00.

971

0.97

40.

984

0.4

0.99

51.

000

0.96

70.

983

0.99

71.

000

0.96

60.

981

0.97

11.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

0.6

1.00

01.

000

0.99

81.

000

1.00

01.

000

1.00

01.

000

0.99

91.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

0.8

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.0

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000



Tab

le6:

Sim

ulat

edpo

wer

sof

prop

osed

test

sfo

rsk

ewed

tdi

stri

buti

onw

ith

sam

ple

size

sm

=n=

50

andm

=n=

100.

cb

m=

n=

50

m=

n=

100

Fis

her

Tip

pet

H-t

est

Fis

her

Tip

pet

H-t

est

T1

T2

T3

T4

T1

T2

T3

T4

T1

T2

T3

T4

T1

T2

T3

T4

1.0

0.0

0.05

30.

045

0.05

10.

053

0.04

80.

045

0.05

10.

048

0.05

30.

051

0.04

70.

048

0.04

80.

049

0.04

80.

051

0.04

70.

048

0.2

0.19

40.

207

0.09

20.

118

0.33

90.

334

0.14

90.

191

0.26

70.

583

0.55

50.

190

0.23

70.

766

0.74

20.

276

0.34

60.

604

0.4

0.88

80.

831

0.43

70.

611

0.96

20.

937

0.57

80.

716

0.80

91.

000

1.00

00.

848

0.89

91.

000

1.00

00.

915

0.94

50.

997

0.6

0.99

90.

996

0.85

50.

942

1.00

00.

998

0.92

00.

969

0.99

21.

000

1.00

00.

998

1.00

01.

000

1.00

01.

000

1.00

01.

000

0.8

1.00

01.

000

0.98

70.

998

1.00

01.

000

0.99

30.

999

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.0

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.2

0.0

0.09

00.

091

0.08

40.

082

0.07

90.

084

0.08

10.

079

0.06

60.

143

0.15

00.

126

0.13

10.

116

0.12

20.

111

0.11

50.

091

0.2

0.25

90.

299

0.14

30.

174

0.38

90.

429

0.19

20.

241

0.30

90.

676

0.70

40.

337

0.38

40.

811

0.82

80.

436

0.49

10.

597

0.4

0.91

50.

880

0.54

80.

687

0.96

60.

958

0.66

10.

782

0.83

11.

000

0.99

90.

902

0.94

71.

000

1.00

00.

952

0.97

00.

997

0.6

1.00

00.

997

0.89

50.

960

1.00

01.

000

0.94

30.

977

0.98

81.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

0.8

1.00

01.

000

0.98

70.

998

1.00

01.

000

0.99

40.

999

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.0

1.00

01.

000

0.99

91.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.4

0.0

0.15

10.

161

0.13

70.

140

0.12

90.

136

0.13

20.

126

0.10

60.

245

0.27

10.

225

0.24

30.

206

0.23

00.

208

0.20

70.

163

0.2

0.34

10.

440

0.23

10.

271

0.47

50.

550

0.28

30.

335

0.37

00.

756

0.82

60.

508

0.55

10.

862

0.90

80.

600

0.64

00.

644

0.4

0.92

20.

920

0.64

10.

728

0.97

50.

974

0.73

70.

824

0.85

01.

000

1.00

00.

946

0.96

51.

000

1.00

00.

975

0.98

50.

994

0.6

0.99

80.

999

0.92

00.

973

1.00

01.

000

0.95

80.

989

0.99

11.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

0.8

1.00

01.

000

0.99

50.

999

1.00

01.

000

0.99

81.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.0

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.6

0.0

0.25

00.

268

0.23

00.

227

0.19

90.

222

0.19

70.

192

0.16

60.

444

0.49

60.

419

0.43

90.

376

0.42

20.

362

0.37

80.

321

0.2

0.43

40.

543

0.31

40.

354

0.53

50.

619

0.34

90.

401

0.45

70.

840

0.90

30.

647

0.68

10.

902

0.94

90.

698

0.73

30.

734

0.4

0.93

90.

944

0.70

30.

779

0.97

90.

977

0.78

00.

858

0.88

61.

000

1.00

00.

971

0.97

61.

000

1.00

00.

986

0.99

00.

998

0.6

1.00

01.

000

0.95

60.

976

1.00

01.

000

0.97

60.

988

0.99

51.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

0.8

1.00

01.

000

0.99

60.

999

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.0

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.8

0.0

0.33

20.

377

0.30

80.

317

0.28

00.

317

0.26

40.

270

0.22

10.

578

0.65

80.

572

0.59

20.

506

0.56

40.

479

0.49

90.

468

0.2

0.48

70.

635

0.37

80.

426

0.55

80.

697

0.38

90.

465

0.50

60.

917

0.95

50.

765

0.79

70.

944

0.97

70.

794

0.82

80.

833

0.4

0.95

50.

966

0.75

60.

817

0.98

30.

987

0.82

70.

875

0.90

61.

000

1.00

00.

986

0.99

31.

000

1.00

00.

993

0.99

80.

999

0.6

0.99

80.

999

0.95

30.

978

0.99

91.

000

0.97

00.

989

0.99

31.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

0.8

1.00

01.

000

0.99

61.

000

1.00

01.

000

0.99

81.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.0

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

2.0

0.0

0.42

70.

486

0.42

90.

422

0.35

80.

414

0.35

30.

362

0.31

50.

722

0.79

00.

706

0.72

80.

629

0.70

80.

618

0.62

90.

625

0.2

0.57

70.

718

0.48

50.

528

0.63

60.

763

0.49

80.

542

0.59

50.

944

0.97

50.

859

0.86

60.

961

0.98

50.

858

0.87

10.

886

0.4

0.95

40.

970

0.79

90.

846

0.97

90.

991

0.85

20.

890

0.92

11.

000

1.00

00.

994

0.99

61.

000

1.00

00.

997

0.99

90.

999

0.6

0.99

80.

999

0.96

80.

982

1.00

01.

000

0.98

40.

992

0.99

41.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

0.8

1.00

01.

000

0.99

71.

000

1.00

01.

000

0.99

81.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.0

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000



Tab

le7:

Sim

ulat

edpo

wer

sof

prop

osed

test

sfo

rno

rmal

dist

ribu

tion

wit

hsa

mpl

esi

zesm

=n=

50

andm

=n=

100.

cb

m=

n=

50

m=

n=

100

Fis

her

Tip

pet

H-t

est

Fis

her

Tip

pet

H-t

est

T1

T2

T3

T4

T1

T2

T3

T4

T1

T2

T3

T4

T1

T2

T3

T4

1.0

0.0

0.05

00.

044

0.04

50.

050

0.04

40.

047

0.04

60.

047

0.04

70.

056

0.05

30.

057

0.05

80.

055

0.05

70.

057

0.05

70.

055

0.2

0.11

70.

145

0.11

10.

120

0.11

30.

158

0.10

70.

097

0.06

60.

211

0.25

70.

216

0.19

40.

219

0.28

70.

199

0.17

70.

066

0.4

0.38

80.

502

0.37

80.

375

0.42

20.

559

0.38

00.

362

0.25

00.

768

0.86

50.

716

0.73

80.

821

0.90

30.

737

0.74

90.

328

0.6

0.77

20.

876

0.74

50.

757

0.81

60.

917

0.74

60.

757

0.65

10.

991

0.99

90.

979

0.98

40.

995

0.99

80.

982

0.98

50.

883

0.8

0.97

50.

993

0.95

80.

968

0.98

30.

997

0.96

10.

971

0.94

51.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

00.

999

1.0

0.99

91.

000

0.99

70.

996

0.99

91.

000

0.99

70.

996

0.99

91.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.2

0.0

0.15

90.

160

0.15

50.

155

0.15

10.

141

0.15

20.

151

0.12

00.

249

0.25

20.

231

0.22

80.

239

0.23

60.

237

0.24

20.

191

0.2

0.22

60.

249

0.21

10.

214

0.18

90.

214

0.18

10.

174

0.13

70.

438

0.49

00.

405

0.42

40.

364

0.42

30.

329

0.33

60.

220

0.4

0.49

60.

582

0.46

90.

496

0.43

80.

564

0.38

50.

407

0.27

10.

833

0.89

80.

797

0.81

60.

799

0.89

40.

714

0.73

70.

430

0.6

0.80

20.

902

0.78

80.

793

0.78

70.

909

0.72

90.

732

0.63

20.

993

1.00

00.

987

0.99

10.

990

0.99

90.

974

0.97

90.

867

0.8

0.96

60.

991

0.95

30.

963

0.96

80.

994

0.94

10.

951

0.92

21.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

00.

997

1.0

0.99

61.

000

0.99

40.

997

0.99

71.

000

0.99

40.

996

0.99

41.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.4

0.0

0.34

50.

346

0.31

80.

323

0.33

70.

323

0.33

40.

336

0.27

00.

564

0.58

30.

548

0.55

20.

574

0.57

20.

578

0.58

00.

510

0.2

0.42

10.

442

0.41

10.

412

0.37

90.

392

0.36

90.

378

0.30

60.

719

0.75

40.

682

0.69

60.

650

0.68

30.

623

0.64

20.

553

0.4

0.65

70.

725

0.62

10.

635

0.54

70.

645

0.51

20.

512

0.44

80.

935

0.95

90.

909

0.91

80.

876

0.93

40.

834

0.84

60.

710

0.6

0.86

80.

926

0.84

10.

858

0.81

00.

906

0.75

30.

754

0.70

30.

998

1.00

00.

991

0.99

60.

994

1.00

00.

979

0.98

50.

920

0.8

0.97

20.

992

0.96

40.

970

0.95

60.

991

0.93

20.

943

0.92

01.

000

1.00

01.

000

1.00

01.

000

1.00

00.

999

1.00

00.

997

1.0

0.99

91.

000

0.99

60.

996

0.99

71.

000

0.99

40.

994

0.99

31.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.6

0.0

0.55

00.

576

0.53

30.

555

0.56

60.

553

0.57

00.

578

0.49

10.

849

0.85

60.

842

0.83

90.

864

0.85

70.

862

0.86

50.

805

0.2

0.63

00.

654

0.60

60.

619

0.58

40.

591

0.57

90.

587

0.51

80.

889

0.90

90.

869

0.87

60.

858

0.87

00.

850

0.85

30.

798

0.4

0.74

10.

798

0.71

90.

725

0.65

00.

721

0.61

10.

617

0.56

20.

980

0.98

60.

963

0.97

10.

945

0.96

90.

917

0.92

30.

862

0.6

0.91

50.

958

0.89

40.

898

0.85

50.

933

0.79

00.

799

0.77

00.

999

1.00

00.

996

0.99

60.

994

1.00

00.

988

0.98

90.

967

0.8

0.98

40.

997

0.97

00.

980

0.97

20.

997

0.92

90.

944

0.94

31.

000

1.00

01.

000

1.00

01.

000

1.00

00.

999

1.00

00.

999

1.0

0.99

91.

000

0.99

70.

999

0.99

81.

000

0.99

10.

993

0.99

11.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.8

0.0

0.70

70.

716

0.68

20.

689

0.71

30.

707

0.71

60.

723

0.63

50.

951

0.95

60.

941

0.94

30.

957

0.95

10.

957

0.95

70.

931

0.2

0.77

10.

794

0.75

90.

763

0.74

50.

749

0.74

70.

752

0.68

20.

964

0.97

60.

961

0.96

60.

966

0.96

50.

959

0.96

30.

933

0.4

0.85

20.

882

0.82

80.

840

0.78

40.

817

0.75

20.

761

0.72

30.

994

0.99

70.

990

0.99

20.

981

0.98

90.

976

0.96

90.

956

0.6

0.95

10.

976

0.92

20.

938

0.90

20.

943

0.84

20.

856

0.85

51.

000

1.00

01.

000

1.00

01.

000

1.00

00.

995

0.99

80.

991

0.8

0.99

10.

997

0.98

30.

986

0.97

10.

990

0.94

50.

951

0.94

81.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.0

0.99

81.

000

1.00

00.

997

0.99

61.

000

0.99

30.

990

0.99

21.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

2.0

0.0

0.83

10.

850

0.81

70.

824

0.85

40.

846

0.85

50.

859

0.77

50.

991

0.99

50.

988

0.99

10.

992

0.99

20.

992

0.99

30.

981

0.2

0.86

50.

879

0.85

90.

856

0.84

90.

854

0.85

20.

853

0.80

00.

991

0.99

20.

986

0.98

90.

991

0.99

00.

992

0.99

00.

979

0.4

0.91

30.

933

0.90

20.

908

0.87

00.

897

0.85

80.

864

0.83

40.

999

1.00

00.

996

0.99

80.

996

0.99

90.

994

0.99

30.

991

0.6

0.96

90.

983

0.95

70.

961

0.93

00.

960

0.90

00.

904

0.90

31.

000

1.00

01.

000

0.99

90.

999

1.00

00.

999

0.99

70.

997

0.8

0.99

50.

999

0.99

00.

992

0.98

20.

991

0.96

00.

967

0.96

71.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.0

0.99

91.

000

0.99

71.

000

0.99

71.

000

0.99

10.

990

0.99

21.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000



Tab

le8:

Sim

ulat

edpo

wer

sof

prop

osed

test

sfo

rt

dist

ribu

tion

wit

hsa

mpl

esi

zesm

=n=

50

andm

=n=

100.

cb

m=

n=

50

m=

n=

100

Fis

her

Tip

pet

H-t

est

Fis

her

Tip

pet

H-t

est

T1

T2

T3

T4

T1

T2

T3

T4

T1

T2

T3

T4

T1

T2

T3

T4

1.0

0.0

0.04

70.

046

0.04

90.

048

0.04

90.

046

0.04

20.

047

0.05

20.

058

0.05

50.

048

0.05

20.

050

0.05

00.

046

0.04

70.

050

0.2

0.08

40.

094

0.09

40.

092

0.09

20.

102

0.09

00.

078

0.05

70.

158

0.17

20.

180

0.16

80.

167

0.18

50.

174

0.15

60.

065

0.4

0.28

00.

309

0.32

70.

309

0.30

40.

345

0.31

20.

278

0.17

00.

596

0.60

30.

614

0.61

80.

641

0.66

20.

626

0.61

70.

215

0.6

0.57

60.

620

0.65

60.

626

0.63

50.

691

0.66

90.

619

0.45

20.

960

0.95

50.

952

0.95

40.

971

0.97

90.

962

0.96

40.

643

0.8

0.85

90.

884

0.89

50.

894

0.90

10.

920

0.90

00.

891

0.80

00.

998

0.99

80.

998

0.99

90.

999

1.00

00.

998

0.99

90.

954

1.0

0.97

40.

982

0.98

60.

980

0.98

Documents

Simultaneously Testing for Location and Scale Parameters of ...1990), majority depth (Singh 1991) and projection depth (Donoho & Gasko 1992). The details about notion of data depth,