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A Jarque-Bera type test for multivariate normality based on second-power skewness and kurtosis

Namhyun Kim1,a

aDepartment of Science, Hongik University, Korea
Correspondence to: 1 Department of Science, Hongik University, 94 Wausan-Ro, Mapo-Gu, Seoul 04066, Korea. E-mail: nhkim@hongik.ac.kr
Received February 12, 2021; Revised April 7, 2021; Accepted May 17, 2021.
Abstract
Desgagné and de Micheaux (2018) proposed an alternative univariate normality test to the Jarque-Bera test. The proposed statistic is based on the sample second power skewness and kurtosis while the Jarque-Bera statistic uses sample Pearson’s skewness and kurtosis that are the third and fourth standardized sample moments, respectively. In this paper, we generalize their statistic to a multivariate version based on orthogonalization or an empirical standardization of data. The proposed multivariate statistic follows chi-squared distribution approximately. A simulation study shows that the proposed statistic has good control of type I error even for a very small sample size when critical values from the approximate distribution are used. It has comparable power to the multivariate version of the Jarque-Bera test with exactly the same idea of the orthogonalization. It also shows much better power for some mixed normal alternatives.
Keywords : Goodness of fit test, Jarque-Bera test, second power kurtosis, second power skewness, multivariate normality, power comparison
1. Introduction

Assessing multivariate normality is an important issue in Statistics because the classical multivariate analysis usually requires the assumption of multinormality. Consequently, there are numerous test procedures devoted to this problem in the literature. Some references for a general review are Henze and Zirkler (1990), Henze (2002), Thode (2002), and Srivastava and Mudholkar (2003). Comparative power study is conducted in Horswell and Looney (1992), Romeu and Ozturk (1993), Mecklin and Mundfrom (2005), Farrell et al. (2007), and Hanusz et al. (2018).

In general, most multivariate tests are generalization of univariate normality tests. In univariate normality tests, the Shapiro andWilk (1965) test and the Jarque and Bera (1980) test are widely used. The Shapiro-Wilk test is a regression and correlation test, and it is known as the most powerful test in the literature (D’Agostino and Stephens, 1986).

Its approximate tests are proposed by DeWet and Venter (1972), Shapiro and Francia (1972). The Shapiro-Wilk and its approximate tests have been generalized to multivariate cases by Malkovich and Afifi (1973), Royston (1983), Fattorini (1986), Srivastava and Hui (1987), Mudholkar et al. (1995), Kim and Bickel (2003), Kim (2004, 2005), and Villasenor-Alva and González-Estrada (2009). Zhou and Shao (2014) proposed a new test related to Fattorini (1986).

The Jarque-Bera test is based on sample Pearson’s skewness and kurtosis, and it is popular especially in econometric fields. A related statistic is proposed by D’Agostino and Pearson (1973, 1974). They used a transformation of D’Agostino (1970) for sample skewness and a transformation of Anscombe and Glynn (1983) for sample kurtosis. Jarque-Bera type tests are generalized to multivariate cases by Doornik and Hansen (2008), Kim (2015, 2016).

Mardia (1970, 1974)’s measure of multivariate skewness and kurtosis is one of the most popular tests for multivariate normality. Mardia and Foster (1983), Kim (2020) studied sum of transformed multivariate skewness and kurtosis. Henze and Zirkler (1990) test is often recommended as a formal test for multinormality because of its invariance and consistency. According to Mecklin and Mundfrom (2005), Farrell et al. (2007), it also has relatively good power over a wide range of alternatives.

Desgagné and de Micheaux (2018) introduced sample second power skewness and kurtosis. And they proposed a statistic on a combination of these measures as an alternative to the Jarque-Bera statistic. The Jarque-Bera test is based on sum of squares of the classical Pearson’s skewness and kurtosis, which are the third and fourth standardized sample moments, respectively. In their terminology, the Jarque-Bera test uses the third power skewness and the fourth power kurtosis. In this paper, we generalize Desgagné and de Micheaux (2018)’s univariate statistic to a multivariate version using orthogonalization or an empirical standardization of data. A simulation is conducted to investigate the size of the test when the approximate distribution of the statistic is used. Also a power comparison is performed through a simulation.

2. Test statistics

Let X1, …, Xn be a sample of independent and identically distributed (i.i.d.) random variables. The Jarque-Bera statistic is based on the classical Pearson’s sample skewness $b1$ and kurtosis b2. The statistic is

$JB=n6(b1)2+n24(b2-3)2.$

Likewise, D’Agostino and Pearson (1973) proposed the statistic

$JBT=(T1 (b1))2+(T2(b2))2,$

to test univariate normality. T1 is a transformation of the null distribution of the skewness $b1$ to normality using a Johnson’s unbounded S U curve proposed by D’Agostino (1970). T2 is a normal approximation for the kurtosis b2 by Anscombe and Glynnn (1983).

Desgagné and de Micheaux (2018) defined the sample second power skewness B2 and the sample second power kurtosis K2. While the Pearson’s skewness $b1$ is the third standardized sample moment

$b1=1n∑i=1nZi3=1n∑i=1n∣Zi∣3sign (Zi),$

they considered the second power skewness B2,

$B2=1n∑i=1n∣Zi∣2sign (Zi),$

where Zi is the standardized observation, Zi = (Xi)/S with $X¯=n-1∑i=1nXi,S2=n-1∑i=1n(Xi-X¯)2$.

The Pearson’s sample kurtosis b2 is the fourth standardized sample moment,

$b2=1n∑i=1n∣Zi∣4.$

Since $n-1∑i=1n∣Zi∣2=1$, they considered the limit $n-1∑i=1n(∣Zi∣2+ɛ-1)/ɛ$ as ε → 0. As a result, the second power kurtosis,

$K2=1n∑i=1nZi2 log∣Zi∣,$

is obtained.

Desgagné and de Micheaux (2018) proposed a normality test statistic to test the null hypothesis H0 : X ~ N(μ,σ) with μ and σ unspecified on a combination of their sample second power skewness and kurtosis defined in (2.4) and (2.6), while the Jarque-Bera statistic in (2.1) uses the third power skewness and the fourth power kurtosis according to their terminology. Their idea is to develop Rao’s score test (Rao, 1948a) on the family of the asymmetric power distribution introduced by Komunjer (2007). They showed the resulting test statistic is a combination of their second power skewness and kurtosis in (2.4) and (2.6).

Desgagné and de Micheaux (2018) showed that the limit distribution of (B2, K2) follows,

$n (-2B2-2-1[K2-(2-log 2-γ)2])→dN2 (0,(4(3-8π)00(3π2-28)32)),$

using the central limit theorem with γ = −Γ′(1)/Γ(1) = 0.577215665 …, Γ(x) is the gamma function. Hence the statistic

$nB223-8π+n(K2-(2-log 2-γ)2)2(3π2-28)8,$

can be used as a test statistic for normality with the asymptotoic chi-squared distribution with 2 degrees of freedom, $χ22$. Note that the above statistic has a very similar form with the Jarque-Bera statistic in (2.1). However, the asymptotic chi-squared distribution is not good enough for small sample sizes. Hence they proposed a modified version of the statistic in (2.7) so that the distribution can be approximated by a chi-squared even for small sample sizes. To do this, they defined the sample second power net kurtosis $K2-B22$ and showed numerically the dependency between B2 and $K2-B22$ is negligible. Next, they found a transformation Z1(B1) and $Z2(K2-B22)$ so that they follow approximately N(0, 1).

Finally, their proposed statistic to test normality based on B2 and K2 in (2.4), (2.6) is

$DX=Z12(B2)+Z22(K2-B22),=nB22(3-8π)(1-1.9n))+n[(K2-B22)13-((2-log 2-γ)2)13 (1-1.026n)]272-1((2-log 2-γ)2)-43(3π2-28)(1-2.25n0.8).$

The statistic DX in (2.8) follows approximately $χ22$, for all n ≥ 10 with high numerical precision.

In this paper, we want to generalize the statistic DX in (2.8) to a multivariate version based on orthogonalization or an empirical standardizaion using the scaled residuals of the observations. Let X1, …, Xn be a p-dimensional i.i.d. random vectors, and let Np(μ, ) be a p-variate multivariate normal distribution with mean vector μ and covariance matrix . We want to test the null hypothesis

$H0:X1,…,Xn is a sample from Np(μ,Σ) for some μ and Σ.$

Let

$X¯=1n∑j=1nXj, S=1n∑j=1n(Xj-X¯) (Xj-X¯)T,$

be a sample mean vector and a sample covariance matrix, respectively. T denotes a transpose. If X1, …, Xn are a sample from Np(μ, ), then Z1, …, Zn with

$Zi=S*T (Xi-X¯), i=1,…,n,$

follow Np(0, I) asymptotically, with 0 the null vector of order p, I the identity matrix of order p × p. Here S* is a matrix satisfying S*TSS* = I. The Zi’s are called the scaled residuals, and each component of Zi = (Z1i, Z2i, …, Zpi), Zki, i = 1, …, n follows approximately independent univariate standard normal N(0, 1) under the null hypothesis. We can construct a multivariate test statistic by adding up the univariate statistic for each coordinate of the scaled residuals in (2.10). The statistic is as follows,

$DXM=∑k=1pDX(k),$

where DX(k) is the statistic in (2.8) for the coordinate (Zk1, Zk2, …, Zkn), k = 1, …, p. The multivariate statistic in (2.11) follows $χ2p2$ approximately.

Malkovich and Afifi (1973) generalized the univariate skewness $b1$ and kurtosis b2 in (2.3) and (2.5) to multivariate cases by Roy’s union-intersection principle (Roy, 1953). That is based on the fact that c′X follows a univariate normal for all c, c ≠ 0, if X follows a multivariate normal. Hence we need to investigate all the possible linear combinations that reduce to a normal under the null hypothesis, and find the direction of giving farthest away from a normal. However, a statistic by that principle is very hard to compute when the dimension of data is large. Hence we need some particular directions to check the normality. Srivastava and Hui (1987) suggested principal component approach for the direction. The statistic in (2.11) is closely related to their idea. Villasenor-Alva and González-Estrada (2009) generalized the Shapiro-Wilk’s test to a multivariate version using the above idea. Kim (2015, 2016) also used the idea to generalize the univariate Jarque-Bera test and related statistics to multivariate cases.

For comparison, we refer to some multivariate normal test statistics. First of all, we consider the multivariate version of the Jarque-Bera type statistics JBM, JBTM proposed by Kim (2016). The statistics are a multivariate generalization of JB in (2.1) and JBT in (2.2) respectively, with exactly the same idea of the orthogonalization. They have the asymptotic distribution $χ2p2$.

Mardia (1970, 1974) defined the multivariate measure of skewness b1,p and kurtosis b2,p as,

$b1,p=1n2∑i=1n∑j=1n{(Xi-X¯)′ S-1 (Xj-X¯)}3,$$b2,p=1n∑j=1n{(Xi-X¯)′ S-1 (Xj-X¯)}2,$

where and S are defined in (2.9). It is known (Mardia, 1970) that,

$ms=nb1,p6→dχ2 (p(p+1)(p+2)6),$$mk=b2,p-p(p+2)8p(p+2)n→dN(0,1).$

Kim (2020) proposed omnibus statistics based on b1,p, b2,p in (2.12), (2.13) for testing multivariate normality using some normalizing transformations. Among them, two statistics MN and NE are considered for comparison, where

$MN=nb1,p6+(b2,p-n-1n+1p(p+2)(n-3)(n-p-1)(n-p+1)(n+1)2(n+3)(n+5)8p(p+2))2,$$NE=Ns2(b1,p)+Ek2(b2,p).$

The transformation is as follows,

$Ns(b1,p)=b1,p-6fn6n2f$$Ek(b2,p)=n8p(p+2)[exp(d(b2,p-p(p+2))-1)d+2p(p+2)(1-2d)n]$

with f = p(p + 1)(p + 2)/6, d = −(p + 8)/(3p(p + 2)). The Ns in (2.18) is a normal approximation of a χ2 variable for the skewness b1,p, which is used in Mardia and Foster (1983). The Ek in (2.19) is a normalizing transformation for the kurtosis b2,p proposed by Enomoto et al. (2019). The statistic MN in (2.16) is almost the same as the statistic Mp by Doornik and Hansen (2008). The statistic MN has the asymptotic distribution $χf+12$, and the statistic NE has $χ2p2$. Kim (2020) recommended NE in (2.17) as an omnibus statistic because of its relatively good power in many alternatives.

As a formal test statistic for multivariate normality, the Henze and Zirkler (1990) statistic Tn,β is frequently recommended. The statistic is as follows.

$Tn,β(X1,…,Xn)=n(4I(S is singular)+Dn,βI(Sis singular)),$

where

$Dn,β=1n2∑i,j=1nexp (-β22‖Yi-Yj‖2)-2(1+β2)-p21n∑j=1nexp (-β22(1+β2)‖Yj‖2)+(1+2β2)-p2$

with,

$‖Yi-Yj‖2=(Xi-Xj)′S-1(Xi-Xj),$

and,

$‖Yj‖2=(Xi-X¯)′S-1(Xj-X¯).$

β is defined as,

$β=βp(n)=12(2p+14)1p+4n1p+4.$

In the next section, we compare the multivariate version of DX statistic, DXM, and the referred ones through a simulation in terms of size of the tests and power.

3. Example and simulation study

### 3.1. Example

We consider the famous data set in Rao (1948b). It consists of the thickness of bark deposit on 28 cork trees measured by the weight of cork borings from the four directions, north (N), east (E), west (W), and south (S). The problem is to investigate if the thickness of bark deposit varies in the four directions. Hence, he selected three constraints to examine if the thickness of bark deposit varies in the four directions,

$Y1=N-E-W+S, Y2=S-W, Y3=N-S,$

by the reason he explained in the paper and tested E(Yi) = 0, i = 1, 2, 3 for this purpose. The assumption of multivariate normality should be valid to test the problem by applying some techniques like Hotelling’s T2 test. Pearson (1956) examined the same dataset.

For the data set (N, E, W, S ) and the contrasts (Y1, Y2, Y3), the statistics DXM in (2.11), ms, mk in the left side of (2.14), (2.15) and the Henze and Zirkler (HZ) statistic in (2.20) are computed in Table 1. The corresponding p-values are also presented. We can use the R-package ‘MVN’ (see Korkmaz et al. (2014)) or ‘mvnormalTest’ for the statistic ms, mk, and HZ.

According to the result, the multivariate normality of the Rao’s original data is rejected at the significance level 0.05 when we use the statistics DXM or HZ. The Mardia’s statistics ms, mk do not reject it. As for the Rao’s data for the contrasts, the multivariate normality cannot be rejected. The data set is also analyzed in Mardia (1975), Srivastava and Hui (1987), and Kim (2015, 2016, 2020).

### 3.2. Comparison of test size

A simulation is performed to compare the proposed DXM in (2.11) with the referred statistics. First of all, we check the test size of the statistics when we use the critical values of the corresponding limit distributions.

As we mentioned, all the statistics DXM, JBM, JBTM, and NE have the limit distribution $χ2p2$ , and MN has χf+1 with f = p(p + 1)(p + 2)/6. The limit distribution of the Henze and Zirkler (1990) statistic in (2.20) is an infinite sum of the weighted $χ12$ variable, and it is omitted for comparison.

For each combination of p = 2, 5 and the sample size n = 10, 20, 30, 40, 50, 100, N = 10, 000 random samples are generated, and the statistics are computed. The samples are from Np(0, I) since the distributions of the statistics do not depend on μ and . Table 2 and Table 3 show the test size of each statistic at the significance level α = 0.05, 0.10. The size is computed by the proportion of the rejections based on the critical values from the limit distribution.

The simulation results indicate that the statistic DXM has a good control of type I error, and satisfies the given significance level very closely for n ≥ 10 when we use the critical values from the approximate distribution $χ2p2$ . The statistics JBM, MN, NE show downward size for small sample sizes, making the tests be conservative. JBTM has a little bigger size. Jönsson (2011) considered the JBM test statistic and suggested a modification to eliminate the size distortion. Apparently, DXM follows the approximate distribution closely even for a small sample size. On the other hand, the other statistics do not have the right size especially for a small sample size.

### 3.3. Power comparison

Next, we compare power of the statistics through a simulation. Tables 47 represent the empirical power of DXM and the referred statistics. The power of JBM, JBTM comes from Kim (2016), and the power of MN, NE and the Henze and Zirkler (HZ) statistic from Kim (2020). A simulation is conducted at the significance level α = 0.05 for the combinations of dimensions p = 2, 5 and sample sizes n = 20, 50. As for DXM, the critical values from the chi-squared distribution are used since Tables 23 indicate that the quantiles from $χ2p2$ give the high accuracy. Except DXM, we used the critical values from the simulation, not from the asymptotic distributions to ensure the true level is α = 0.05. Regarding the HZ statistic, we used the critical values given in Table 3.1 and 3.3 in Henze and Zirkler (1990).

We generate N = 5,000 samples from each of the various alternative distributions. The alternatives in Kim (2020) are selected for comparison. They are the distributions with independent marginals and mixtures of normal distributions. F1 * F2 denotes the distribution with independent marginal distributions F1 and F2. The product of p independent copies of F1 is denoted $F1p$. NMIXp(κ, μ, ρ1, ρ2) stands for the normal mixture

$κNp(0,R1)+(1-κ)Np(μ1,R2),$

where Ri is a matrix with diagonal elements equal to 1 and off-diagonal equal to ρi, 0 ≤ ρi < 1, i = 1, 2. According to Mecklin and Mundfrom (2005), the mixture of normal with κ = 0.5 is severely contaminated, symmetric and platykurtic (short-tailed), and the mixture with κ = 0.9 is mildly contaminated, skewed and leptokurtic (long-tailed).

The empirical power analysis shows the following. The remarkably best power is written in bold for distinction. First, all the statistics considered show similar behavior in power. They show better power against alternatives with symmetric thicker tailed marginals such as Cauchy and t-distributions than those with symmetric thinner tailed beta marginals. They show relatively good power against skewed marginal alternatives. The HZ statistic has the best power for alternatives with skewed marginals when p = 2, n = 20. However, this phenomena becomes weak when the sample size and the dimension of the data are bigger. Usually most test statistics for multivariate normality show extremely poor power against beta marginal alternatives. As Kim (2020) pointed out, the statistic NE shows relatively good power against symmetric light tailed marginal alternatives. The statistic HZ also has good power for these alternatives. The power of DXM is lower than NE or HZ against the beta marginal alternatives. However, it is as good as JBM or JBTM.

Second, the proposed statistic DXM shows outstanding power against some normal mixtures, where the contaminating distribution in (3.1) has a mean vector away from 0. Kim (2020) mentioned that the statistic MN has relatively good power against the normal mixture alternatives among the omnibus test statistics based on Mardia’s skewness and kurtosis. The HZ statistic also shows good power for these alternatives. In the meantime, the DXM statistic shows much better power than these statistics. Third, the statistic DXM shows better or comparable power against almost all the alternatives considered, when we compare DXM with JBM, JBTM. The two statistics JBM, JBTM are counterpart to DXM in that they are based on the classical skewness and kurtosis, and they are generalized to a multivariate version by the exactly same way.

4. Conclusions

Desgagné and de Micheaux (2018) defined the sample second power skewness and kurtosis, and proposed a normality test statistic based on them. They suggested it as an alternative to the Jarque-Beta statistic, which is based on the classical Pearson’s sample skewness and kurtosis. In this paper, we generalized the univariate normality test statistic proposed by Desgagné and de Micheaux (2018) to a multivariate version based on orthogonalization.

A simulation study reveals that the proposed statistic has the right test size even for a small sample size when we use the critical values from the approximate distribution, $χ2p2$. It has very good control of type I error compared to the other referred statistics. So p-values of the test can be computed with $χ2p2$ distribution with high precision; we need no tables or simulated quantiles. In addition, the statistic has relatively good power against alternatives considered in the study, and shows the best power against some mixtures of multivariate normal distribution.

TABLES

### Table 1

Statistics and p-values for the Rao’s bark deposit data

(N, E, W, S)(Y1, Y2, Y3)

DXMmsmkHZDXMmsmkHZ
Statistics17.709020.890−0.3980.99908.1815.493−0.6960.679

p-value0.02350.4040.6900.01110.2250.8560.4860.276

### Table 2

Test size using the approximate distribution for p = 2

Statisticαn = 10n = 20n = 30n = 40n = 50n = 100
DXM0.050.05460.05150.05380.05360.05200.0488
JBM0.050.00780.02840.04020.04260.04800.0544
JBTM0.050.06780.06560.06200.06000.07020.0648
MN0.050.03170.04900.05070.05670.05490.0553
NE0.050.00590.03250.03610.03980.04480.0507

DXM0.100.10970.09750.10270.09980.10090.0960
JBM0.100.01580.04360.05320.06080.06980.0800
JBTM0.100.10560.10360.10580.10040.11480.1116
MN0.100.05060.07130.07650.08610.08450.0896
NE0.100.01820.05090.05790.06210.06690.0777

### Table 3

Test size using the approximate distribution for p = 5

Statisticαn = 10n = 20n = 30n = 40n = 50n = 100
DXM0.050.05800.04720.05340.05120.05400.0526
JBM0.050.00200.02880.04160.03840.05360.0658
JBTM0.050.07240.07460.07100.07080.07040.0584
MN0.050.00200.02150.03640.04220.04470.0506
NE0.050.02860.04230.04810.05300.05740.0603

DXM0.100.11550.09290.10080.10050.10450.1017
JBM0.100.00520.03980.06180.05660.07600.0938
JBTM0.100.11060.11980.11960.11960.11980.1062
MN0.100.00420.03650.05790.06590.07150.0907
NE0.100.09170.08940.09540.09840.10280.1050

### Table 4

Power comparison of the statistics (α = 0.05, p = 2, n = 20)

AlternativeDXMJBMJBTMMNNEHZ
N(0, 1)20.050.040.040.050.060.05
Cauchy(0, 1)20.970.940.960.970.940.97
Logistic(0, 1)20.140.160.150.170.140.10
(t2)20.710.720.680.740.670.64
(t5)20.250.250.250.300.250.17

Beta(1, 1)20.100.000.060.000.230.18
Beta(2, 2)20.040.010.020.000.080.06

Beta(1, 2)20.150.050.060.040.090.28
exp(1)20.730.660.640.690.710.86
Lognormal(0, 0.5)20.510.480.500.500.510.58
Gamma(0.5, 1)20.920.870.850.900.930.99
Gamma(5, 1)20.210.200.230.210.220.24
$(χ52)2$0.380.340.370.370.380.45
$(χ152)2$0.140.180.160.160.160.16

N(0, 1) * t50.160.170.160.170.150.12
N(0, 1) * Beta(1, 1)0.110.020.070.010.070.10
N(0, 1) * exp(1)0.370.360.350.400.400.51
$N(0,1)*χ52$0.270.240.260.200.190.22

NMIX2(0.5, 4, 0, 0)0.890.040.490.010.150.52
NMIX2(0.5, 0, 0, 0.9)0.190.170.180.180.150.14
NMIX2(0.5, 4, 0, 0.9)0.820.220.500.230.270.82
NMIX2(0.9, 4, 0, 0)0.730.660.610.550.540.64
NMIX2(0.9, 0, 0, 0.9)0.060.070.070.060.060.05
NMIX2(0.9, 4, 0, 0.9)0.710.650.650.540.570.62

### Table 5

Power comparison of the statistics (α = 0.05, p = 2, n = 50)

AlternativeDXMJBMJBTMMNNEHZ
N(0, 1)20.050.040.050.050.050.05
Cauchy(0, 1)21.001.001.001.001.001.00
Logistic(0, 1)20.250.280.270.320.260.15
(t2)20.970.940.940.970.940.95
(t5)20.500.510.490.560.480.32

Beta(1, 1)20.320.000.370.000.810.68
Beta(2, 2)20.100.000.100.000.310.17

Beta(1, 2)20.450.100.200.120.290.80
exp(1)20.980.940.961.001.001.00
Lognormal(0, 0.5)20.900.850.840.950.950.94
Gamma(0.5, 1)21.000.990.991.001.001.00
Gamma(5, 1)20.530.460.480.560.610.53
$(χ52)2$0.800.740.750.860.880.87
$(χ152)2$0.390.330.360.410.450.36

N(0, 1) * t50.320.340.300.320.280.17
N(0, 1) * Beta(1, 1)0.430.030.540.010.170.32
N(0, 1) * exp(1)0.800.730.730.880.900.92
$N(0,1)*χ52$0.720.590.590.540.560.52

NMIX2(0.5, 4, 0, 0)1.000.080.990.010.361.00
NMIX2(0.5, 0, 0, 0.9)0.480.410.330.280.210.32
NMIX2(0.5, 4, 0, 0.9)1.000.550.980.900.931.00
NMIX2(0.9, 4, 0, 0)0.990.990.980.970.980.95
NMIX2(0.9, 0, 0, 0.9)0.070.060.090.080.090.05
NMIX2(0.9, 4, 0, 0.9)0.990.980.980.980.980.94

### Table 6

Power comparison of the statistics (α = 0.05, p = 5, n = 20)

AlternativeDXMJBMJBTMMNNEHZ
N(0, 1)20.050.050.040.050.050.04
Cauchy(0, 1)21.000.990.991.000.990.99
Logistic(0, 1)20.120.140.140.180.070.08
(t2)20.840.840.850.880.780.69
(t5)20.260.280.280.340.160.14

Beta(1, 1)20.050.020.030.000.290.10
Beta(2, 2)20.040.020.020.000.160.06

Beta(1, 2)20.080.050.060.030.080.18
exp(1)20.660.610.640.790.580.81
Lognormal(0, 0.5)20.490.490.490.600.390.49
Gamma(0.5, 1)20.910.860.880.970.910.99
Gamma(5, 1)20.170.170.160.210.090.17
$(χ52)2$0.300.300.350.390.200.34
$(χ152)2$0.120.140.140.150.060.12

N(0, 1) * t50.100.120.120.100.060.06
N(0, 1) * Beta(1, 1)0.050.030.040.030.070.06
N(0, 1) * exp(1)0.140.170.150.180.080.13
$N(0,1)*χ52$0.190.180.170.110.050.08

NMIX5(0.5, 4, 0, 0)1.000.040.720.030.090.17
NMIX5(0.5, 0, 0, 0.9)0.520.430.440.650.410.54
NMIX5(0.5, 4, 0, 0.9)0.950.390.650.820.650.96
NMIX5(0.9, 4, 0, 0)0.860.700.660.400.240.32
NMIX5(0.9, 0, 0, 0.9)0.090.100.090.100.050.07
NMIX5(0.9, 4, 0, 0.9)0.840.750.700.430.220.40

### Table 7

Power comparison of the statistics (α = 0.05, p = 5, n = 50)

AlternativeDXMJBMJBTMMNNEHZ
N(0, 1)20.050.050.060.040.050.04
Cauchy(0, 1)21.001.001.001.001.001.00
Logistic(0, 1)20.240.260.240.410.320.14
(t2)21.000.990.991.001.001.00
(t5)20.550.560.540.680.670.30

Beta(1, 1)20.070.010.100.000.830.50
Beta(2, 2)20.050.010.040.000.420.14

Beta(1, 2)20.190.030.080.050.110.66
exp(1)20.980.950.961.001.001.00
Lognormal(0, 0.5)20.890.870.890.990.990.96
Gamma(0.5, 1)21.000.991.001.001.001.00
Gamma(5, 1)20.440.430.440.670.550.47
$(χ52)2$0.750.690.690.950.920.88
$(χ152)2$0.300.330.320.490.360.28

N(0, 1) * t50.220.250.210.200.150.09
N(0, 1) * Beta(1, 1)0.150.060.240.020.070.10
N(0, 1) * exp(1)0.350.350.370.550.440.39
$N(0,1)*χ52$0.540.420.460.250.200.16

NMIX5(0.5, 4, 0, 0)1.000.111.000.030.110.70
NMIX5(0.5, 0, 0, 0.9)0.960.840.800.900.860.98
NMIX5(0.5, 4, 0, 0.9)1.000.860.991.001.001.00
NMIX5(0.9, 4, 0, 0)1.000.980.970.880.810.92
NMIX5(0.9, 0, 0, 0.9)0.160.150.150.170.120.11
NMIX5(0.9, 4, 0, 0.9)0.991.001.000.990.960.91

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