A New Multivariate Kurtosis and Its Asymptotic Distribution

A ew Multivariate Kurtosis and Its Asymptotic Distribution Chiaki Miyagawa 1 and Takashi Seo 1 Department of Mathematical Information Science, Graduate School of Science, Tokyo University of Science, Tokyo, Japan Department of Mathematical Information Science, Faculty of Science, Tokyo University of Science, Tokyo, Japan Abstract In this paper, we propose a new definition for multivariate kurtosis based on the two measures of multivariate kurtosis defined by Mardia 1970 and Srivastava 1984, respectively. Under normality, the exact values of the expectation and the variance for the new multivariate sample measure of kurtosis are given. We also give the third moments for the sample measure of new multivariate kurtosis. After that standardized statistics and normalizing transformation statistic for the sample measure of a new multivariate kurtosis are derived by using these results. Finally, in order to evaluate accuracy of these statistics, we present the numerical results by Monte Carlo simulation for some selected values of parameters. Keywords: Asymptotic distribution, Measure of multivariate kurtosis, Multivariate normal distribution, ormalizing transformation statistic, Standardized statistics. 1 Introduction In multivariate statistical analysis, the test for multivariate normality is an important problem. This problem has been considered by many authors. Shapiro and Wilk 1965 derived test statistic, which is well known as the univariate normality test. This Shapiro-Wilk test were extend for the multivariate case by Malkovich and Afifi 1973, Royston 1983, Srivastava and Hui 1987 and so on. Small 1980 gave multivariate extensions of univariate skewness and kurtosis. For a comparison of these methods, see, Looney 1995. To assess multivariate normality, the multivariate sample measures of skewness and kurtosis have been defined and their null distributions have been given in Mardia 1970, 1974. Srivastava 1984 also has proposed another definition for the sample measures of multivariate skewness and kurtosis and their asymptotic distributions. Recently, Song 001 has given a definition which is different from Mardia s and Srivastava s measure of multivariate kurtosis. Srivastava s sample measures of multivariate skewness and kurtosis have been discussed by many authors. Seo and Ariga 009 derived the normalizing transformation statistic for Srivastava s sample measure of multivariate kurtosis and its asymptotic distribution. Okamoto and Seo 010 derived the exact 1

values of the expectation and the variance for a sample measure of Srivastava s skewness and improved approximate χ test statistic for assessing multivariate normality. On the other hand, Jarque and Bera 1987 proposed the bivariate test using skewness and kurtosis for univariate case. The improved Jarque-Bera test statistics have been considered by many authors. Mardia and Foster 1983 proposed test statistic using Mardia s sample measures of skewness and kurtosis. Koizumi, Okamoto and Seo 009 proposed multivariate Jarque- Bera test statistic using Mardia s and Srivastava s skewness and kurtosis. Recently, Enomoto, Okamoto and Seo 010 gave a new multivariate normality test statistic using Srivastava s skewness and kurtosis. In this way, it is many studies which the problem for multivariate normality test has been discussed by using skewness and kurtosis. We focus on multivariate kurtosis in this paper. Our purposes are to propose a new definition of multivariate kurtosis from the definition in Mardia 1970 and Srivastava 1984 and to give the asymptotic distribution. In order to achieve our purposes, we derive the first, second and third moments for a sample measure of multivariate kurtosis under multivariate normality where the population covariance matrix Σ is known. Further we give the standardized statistics and the normalizing transformation statistic. Finally, we investigate the accuracy of the expectations, the variances, the skewnesses, the kurtosises and the upper percentile for these statistics by Monte Carlo simulation for some selected parameters. Some definitions of multivariate kurtosis.1 Mardia s measure of multivariate kurtosis First, we discuss a measure of multivariate kurtosis defined by Mardia 1970. Let x be a random p-vector with the mean vector µ and the covariance matrix Σ. Then Mardia 1970 has defined the population measure of multivariate kurtosis as β M = E {x µ Σ 1 x µ } ]. Then we can write β M = E {trz } ], 1 where z = z 1,..., z p = Σ 1/ x µ and Z = diagz 1, z..., z p. We note that β M = pp under multivariate normality. Let x 1,..., x be sample observation vectors of size from a multivariate population. Let x = 1 x and S = 1 x xx x be the sample mean vector and the sample covariance matrix based on sample size, respectively. Then the sample measure of multivariate kurtosis in Mardia 1970 is defined as b M = 1 { x x S 1 x x }. Further Mardia 1970 has obtained asymptotic distributions of b M used them to test the multivariate normality. For the moments and approximation to the null distribution of Mardia s measure of multivariate kurtosis, see, Mardia and Kanazawa 1983, Siotani, Hayakawa and

Fujikoshi 1985. Theorem 1 Mardia 1970. Let b M be the sample measure of multivariate kurtosis on the basis of random samples of size drawn from p µ, Σ where Σ is unknown. Then is asymptotically distributed as 0, 1. z M = b M pp {8pp /} 1/. Srivastava s measure of multivariate kurtosis ext, we consider Srivastava s measure of multivariate kurtosis which is different from the definition by Mardia 1970. Srivastava 1984 gave a definition for a measure of kurtosis for multivariate populations using the principle component method. Let x be a random p- vector with the mean vector µ and the covariance matrix Σ. Let Γ = γ 1, γ,..., γ p be an orthogonal matrix such that Σ = ΓD λ Γ, where D λ = diagλ 1, λ,..., λ p and λ 1,..., λ p are the characteristic roots of Σ. Then Srivastava 1984 defined the population measure of multivariate kurtosis as β S = 1 p p i=1 Ey i θ i 4 ], λ i where y i = γ ix and θ i = γ iµ, i = 1,,..., p. Therefore we can write β S = 1 p E trz 4]. We note that β S = 3 under multivariate normality. For the moments and approximation to the null distribution of Srivastava s measure of multivariate kurtosis, see, Seo and Ariga 006. Let x 1,..., x be samples of size from a multivariate population. Let x and S = HD ω H be the sample mean vector and the sample covariance matrix based on sample size, where H = h 1, h,..., h p is an orthogonal matrix and D ω = diagω 1, ω,..., ω p. We note that ω 1, ω,..., ω p are the characteristic roots of S. Then the sample measure of multivariate kurtosis in Srivastava 1984 is defined as b S = 1 p p i=1 1 ω i y i y i 4. Further Srivastava 1984 has obtained asymptotic distributions of b S used them to test the multivariate normality. Theorem Srivastava 1984. Let b S be the sample measure of multivariate kurtosis on the basis of random samples of size drawn from p µ, Σ where Σ is unknown. Then p z S = 4 b s 3 is asymptotically distributed as 0, 1. 3

3 A new measure of multivariate kurtosis From Mardia 1970, Srivastava 1984, two measures of multivariate kurtosis are based on forth moments. So we propose a new measure of multivariate kurtosis from Mardia s definition and Srivastava s that. 3.1 A new measure of multivariate kurtosis for multivariate populations Let x be a random p-vecter with the mean vector µ and the covariance matrix Σ. From 1 and, we propose that β MS = 1 p E {trz} 4]. Let Γ = γ 1, γ,..., γ p be an orthogonal matrix such that Σ = ΓD λ Γ, where D λ = diagλ 1, λ,..., λ p and λ 1,..., λ p are the characteristic roots of Σ. Therefore we can defined as β MS = 1 p E p {trz} 4] = 1 4 p E y i θ i, λi where, y i = γ ix and θ i = γ iµ, i = 1,,..., p. We note that β MS under multivariate normality. i=1 3. A sample measure of the new multivariate kurtosis Let x 1, x,..., x be p-dimensional sample vectors of size from a multivariate population. In addition, let x and S = HD ω H be the sample mean vector and the sample covariance matrix, where H = h 1, h,..., h p is an orthogonal matrix and D ω = diagω 1, ω,..., ω p. We note that ω 1, ω,..., ω p are the characteristic roots of S. Then a new sample measure of multivariate kurtosis is defined as b MS = 1 p p i=1 y i y i ωi 4. Without loss of generality, we may assume that Σ = I and µ = 0 when we consider this sample measure of multivariate kurtosis. In this paper, we consider the moments for the case when Σ is known under normality. Since we can write λ i = 1i = 1,,..., p in this case, we can reduce b MS to as follows; b MS = 1 p { p 4 y i y i }. i=1 4

4 First moment of b MS We consider the expectation of b MS under multivariate normality. Eb MS ] given by { Eb MS ] = E 1 p } 4 y p i y i=1 First we can expand = 1 p EA4 i] A 4 p p 1EA3 ia j ] B 3 p p 1EA ia j] C 6 p p 1p EA ia j A k ] D 1 p p 1p p 3EA ia j A k A l ] E, 3 where A i = y i y. In order to avoid the dependence of y i and y i, let y i defined on the subset of y i1, y i,..., y i by deleting y i, that is, be a mean y i = 1 1 j=1,j Putting y i = z/ 1, we have A v i = y i y i v = 1 1 v v y i y i = 1 1 y ij. v y i Then we note that the odd order moments of z and y i equal zero and For the case of v = 1, 3, 5,..., we have For the case of v =, 4, 6,..., 1, we have Ez k ] = Ey k i] = k 1 5 3 1, k = 1,,.... E A v i] = 0. E ] A 1 i =, E ] A 4 3 1 i = E ] A 8 105 1 4 i =, E A 10 4 i, E ] A 6 15 1 3 i =, 3 ] 945 1 5 =, E A 1 5 i Calculating the cases of A E in 3 with respect to y i and z; z 1 v. ] = 10395 1 6 6. A EA 4 3 1 i] =, B EA 3 ia j ] = 0, C EA ia j] = D EA ia j A k ] = 0, E EA ia j A k A l ] = 0, 1, we obtain Eb MS ] = 3 6 3. 4 5

5 Variance of b MS In this section, we consider the variance of b MS. To obtain Var b MS ], we expand E b MS ] as follows. E { ] b 1 p } 4 MS = p E y 4 i y i = 1 p E 4 B 1 B B 3 B 1 i=1 B 3 B 3 B 4 B 5 B 1 B B 3 B 4 B 1 B 4 B 5 B B 1 B B 4 B 5 B 5 ] B 5, where B 1 = B 4 = p i=1 p A 4 i, B = p i j 1A ia j A k, B 5 = 4A 3 ia j, B 3 = p i<j p A i A j A k A l, i,j,l,l 6A ia j, and A i = y i y i. In order to avoid the dependence of y i, y iβ, and y i in EA i A iβ ], let y,β i be a mean defined on the subset y i1, y i,..., y i by deleting y i and y iβ, that is, Putting y,β i y,β i = 1 j=1,j, β = z/, where z 0, 1, we have y ij. = = A u ia v iβ = y i y i u y iβ y i v { 1 1 y i y,β i 1 y iβ { 1 1 y i z 1 y iβ } u { 1 1 } u { 1 1 y iβ y iβ y,β i 1 y i z 1 } v y i } v 6

If the value of u is odd and that of v is even, then E A u ia v iβ] = 0. Otherwise, for example, after a great deal of calculation for the expectations, we get E A i A iβ ] = 1, E ] A ia 3 iβ =, E ] A 3 3 1 ia iβ =, E ] A 3 ia 3 33 6 5 iβ =, E ] A 4 ia 3 1 5 3 iβ =, 3 E ] A 4 ia 4 33 4 1 3 4 60 35 iβ =, E ] A 5 15 1 ia 4 iβ =, 3 E ] A 5 ia 3 15 13 6 7 iβ =, E ] A 6 ia 15 1 7 4 iβ =, 4 E ] A 6 ia 4 45 1 4 4 3 8 8 1 iβ =, 5 E ] A 7 ia 3 315 1 3 iβ =, E ] A 8 ia 105 1 3 9 5 iβ =, 5 E ] A 8 ia 4 315 1 4 4 3 4 18 33 iβ =. 6 By using the above results, the expectation for the each term of Eb MS ] is given by as follows. E = E p Ai j 4 pp 1 A 4 B 1 33 4 0 3 86 108 45 9 1 4 = p pp 1 A 4 i = 3p 1{3 3 9p 3 9p 8 3p 48}, E B = E p i j p = E 4A 3 ia j i j 4A 3 ia j p 4A 3 ia j 4A 3 ia k p 4A 3 ia j 4A 3 ja i i j p 4A 3 ia k 4A 3 ja k = 96pp 14 3 13 15 6, p 4A 3 ia k 4A 3 ja i p ] 4A 3 ia k 4A 3 ja l i.j,k,l 7

E B 3 = E p i<j p = E 6A iaj i<j p i<j<k<l 6A ia j 6A ia j p i<j<k 6A ka l 6A ia j ] 6A ia k = 9p 1{ 3 pp 1 p3p 11 3p 19p 8 9p 16}, E B 4 =E p p = E 1A ia j A k 1A ia j A k p 1A ia j A k 1A ia j A l,l p 1A ia j A k 1A ja i A k p 1A ia k A l 1A ja i A k i.j,k,l p,l,m p,l,m p i.j,k,l,m,n p 1A ia j A k 1A ja i A l,l p,l,m 1A ia k A l 1A ja i A m 1A ia k A l 1A ja k A m 1A ia j A k 1A ia l A m p 1A ia k A l 1A ja k A l i.j,k,l ] 1A ia k A l 1A ja m A n = 7 1pp 1p { p p 1 p 4}, 8

E E E B 5 = E p,l p = E 4A i A j A k A l,l p,l,m,n p,l,m,n,o p,l,m,n,o,q 4A i A j A k A l p,l,m 4A i A j A k A l 4A i A j A k A m 4A i A j A k A l 4A i A j A m A n 4A i A j A k A l 4A i A m A n A o 4A i A j A k A l 4A m A n A o A q = 4pp 1p p 3 3 4 6 3, B 1 { p = E = 0, i j p i j B 1 A 4 i B A 4 i ] = E 4A 3 ia j 4A 3 ja i { p = E i j B 3 A 4 i ] = E p i=1 6A ia j = 18pp 1 13 { 1p 8}, A 4 i p p i=1 A 4 i A 4 i i j ] p 4A 3 ia j }] 4A 3 ja k p p i<j A 4 i 6A ia j ] 6A ja k }] 9

E E E B 1 { p = E = 0, i j B 1 { p = E = 0, p,l,l,m B B 4 A 4 i A 4 i ] = E p i=1 A 4 i 1A ia j A k B 5 A 4 i ] p,l = E A 4 i i=1 p ] p 1A ia j A k A 4 i 1A ja i A k }] 1A ja k A l p A 4 i 4A i A j A k A l }] 4A j A k A l A m B 3 ] = E { p = E 4A 3 ia j = 0, i j p 4A 3 ia j i j 6A ia j 6A ja k,l p 4A 3 ia j ] p 4A i A j A k A l p i<j 6A ia j p 4A 3 ia k p 4A 3 ia l,l ] 6A ia j 6A ja k }] 10

E E B B 4 ] = E i j { p = E 4A 3 ia j 1A ia j A k = p 4A 3 ia j 1A ja i A k p 4A 3 ia j 1A ja k A l,l p 4A 3 ia j 1A ka j A l,l 88pp 1p 13, B B 5 ] = E i j p 4A 3 ia j ] p 1A ia j A k p 4A 3 ia j 1A ia k A l,l p 4A 3 ia j 1A ka i A j p 4A 3 ia j 1A ka i A l,l p,l,m { p = E 4A 3 ia j 4A i A j A k A l = 0, p,l,l,m p,l,m p,l,m,n 4A 3 ia j 4A i A k A l A m 4A 3 ia j 4A j A k A l A m }] 4A 3 ia j 1A ka l A m p 4A 3 ia j }] 4A 3 ia j 4A k A l A m A n,l ] p 4A i A j A k A l 11

E E E B 3 { p = E = 0, p p,l,m B 3 B 4 6A ia j 6A ia j { p = E = 0,,l p,l,m,n B 4 6A ia j ] = E p i<j 1A ia j A k 1A ka i A j B 5 6A ia j 6A ia j B 5 6A ia j p,l p,l }] 1A ka l A m ] = E p i<j 6A ia j 4A i A j A k A l 6A ia j 6A ia j p,l,m }] 4A k A l A m A n ] =E ] p 1A ia j A k 1A ia k A l 1A ka i A l,l p 1A ia j A k { p = E 1A ia j A k 4A i A j A k A l = 0. p,l,l,m p,l,m,n p,l,m p,l,m,n p,l,m,n,o 1A ia j A k 4A i A j A l A m 1A ia j A k 4A i A l A m A n 1A ia j A k 4A j A k A l A m 1A ia j A k 4A j A l A m A n ] p 4A i A j A k A l 6A ia j,l }] 1A ia j A k 4A l A m A n A o 1 4A i A k A l A m ] p 4A i A j A k A l

Therefore Hence we get E ] b 60 MS = 9 58 34 135 3. 4 Var b MS ] = 96 31 360 3 144 4. 5 6 Third moment of b MS In this section, we consider E b 3 MS ] in order to obtain normalizing transformation statistic. As for the normalizing transformation statistic, we discuss in Section 7. ow we can expand Eb 3 MS ] given by E ] b 3 MS = E 1 p 4 3 y p i y i i=1 3 1 = 3 p E χ 4 6 = p 1 p E 6 i=1 1 p 6 1 A i p E 3 1 p 6 p E p A i8 i=1 i=1 4 A iγ, i=1 A i4 p i=1 A iβ4 p i=1 4 A iβ where χ = p ı=1 y i y i = p i=1 A i. In order to avoid the dependence of y i, y iβ, y iγ and y i, let y,β,γ i be a mean defined on the subset of y i1, y i,..., y i by deleting y i, y iβ and y iγ, that is, y,β,γ i = 1 y ij. 3 Putting y,β,γ i = z/ 3, we have E y i y i u y iβ y i v y iγ y i w ] j=1, j,β,γ { = E 1 1 3 y i 3 { 1 1 y iβ { 1 1 y iγ z 1 y iβ 1 y γ z 1 y iγ 1 y 3 z 1 y i 1 y β } u } v } w ]. If the values of u, v and w are odd, even and even, respectively, or if all of them are odd, then we have E A u ia v iβ Aw iγ] = 0. Otherwise, for example, after a great deal of calculation for the 13

expectations, we obtain E ] A i A iβ A 1 iγ = 3 O 3, E ] A i A iβ A 4 3 iγ = 18 O 3, E ] A ia iβa 3 iγ = 1 9 O 3, E ] A ia iβa 4 1 iγ = 3 48 O 3, E ] A ia 3 iβa 3 9 iγ = 45 O 3, E ] A ia 4 iβa 4 45 iγ = 9 34 O 3, E ] A 3 ia 4 iβa 4 7 iγ = 1 16 O 3, E ] A 4 ia 4 iβa 4 16 iγ = 7 1053 O 3. Therefore the expectation for the each term of Eb 3 MS ] is given by as follows. p 1 1 p E A 6 i = 10395 O 3, i=1 p p 4 3 1 E A i8 A p 6 iβ = 945 6615 O 3, i=1 i=1 p p p 4 1 E A i4 A iβ4 A p 6 iγ i=1 i=1 i=1 = 7 43 71p4 7p 3 51p 40p 108 p O 3. Summarizing these results, we get E ] b 3 70 MS = 7 71p4 7p 3 391p 40p 108 O 3. 6 p 7 Standardized Statistics and normalizing transformation statistic for Eb MS ] By using the results of the expectation and the variance for the sample measures of multivariate kurtosis, we obtain as following theorem. Theorem 3 Let x 1, x,..., x be random samples of size drawn from p µ, Σ, where Σ is known. Then z MS = z MS = p 4 b MS 3, 4 4 4 3 13 15 6 { b MS 3 6 3 } are asymptotically distributed as 0, 1. ext an asymptotic expansion of the distribution function for a new sample measure of multivariate kurtosis b MS is given under the multivariate normal population. Further, as an 14

improved approximation to a standard normal distribution, we derive the normalizing transformation for the distribution of b MS β MS. Let Y MS = b MS β MS. Then we have the following distribution function for b MS. Pr bms β MS σ y ] = Φy 1 { a1 σ Φ1 y a } 3 σ 3 Φ3 y O 1, where Φy is the cumulative distribution function of 0, 1 and Φ j y is the jth derivation of Φy. Also a 1, σ and a 3 are coefficients for the first three cumulants of Y MS taken the following form; κ 1 Y MS = a 1 O 3, where κ Y MS = σ O 1, κ 3 Y MS = 6a 3 O 3, a 1 = 6, σ = 96, a 3 = 1584 p. Further we put the function gb MS satisfying the following equation. a 3 σ σg b MS 3 g b MS = 0 where g β S 0. Solving this equation, we have gb MS = 3/11exp 11/3b MS ]. Therefore the above distribution function is transformed as ] {gbms gβ MS c/} Pr y = Φy O 1, σ where c = 45/exp 11/3b MS ] Hence we have a following theorem. Theorem 4 Let x 1, x,..., x be random samples of size drawn from p µ, Σ, where Σ is known. Then { 11 3 z T = exp 11b ] 3 MS 3 exp ] } 33 11 3 c/ ] 96exp 33 3 is normalizing transformation for b MS, where c = 45/exp 33/3b MS ]. For the normalizing transformation of some statistics in multivariate analysis, see, Konishi 1981, Seo, Kanda and Fujikoshi 1994 and so on. 15

8 Simulation studies We investigate the accuracy of standardized statistics z MS, zms and normalizing transformation statistic z T by Monte Carlo simulation. Parameters of the dimension and the sample size in simulation are as follows: p = 3, 5, 7, 10, = 0, 50, 100, 00, 400, 800. As a numerical experiment, we carry out 1, 000, 000 replications for the case where Σ= I is known. Table 1 gives the values of the expectation and the variance for z MS, zms and z T. LT s in Table 1 denote the limiting term for the expectation and the variance of a new multivariate kurtosis. Table gives the values of the skewness and the kurtosis for z MS, zms and z T. LT s in Table denote the limiting term for the skewness and the kurtosis of a new multivariate kurtosis. From Tables 1 and, it may be noted that the values for each statistic give good normal approximations as is large. It may be seen from Table 1 that the expectation and the variance of all statistics converge to zero and one, as is large. The results show that the theorems 3 and 4 hold. Particularly, the expectations and the variances of the statistic zms are almost same for any. That is, z MS is almost close to the limiting term even for small, respectively, since zms is an standardized statistic using the exact values of the expectation and the variance derived in this paper. As for the expectation, the accuracy of approximation for z T is better than that for z MS for any. Hence, it may be noticed that both zms and z T are improvement statistics of z MS. It may be seen from Table 1 that there is not the effect of dimension at all. We note that the value of skewness is zero and the value of kurtosis is three under standard normal distribution. It may be seen from Table that all statistics converge to zero and three as is large. z MS and zms are the same values since they are improved statistics for the expectation and the variance. On the other hand, from Theorem 4, we note that z T is improved for the distribution function. Therefore it may be noted from Table that the values of skewness and kurtosis for z T rapidly converge to zero and three. Further it may be seen that the normalizing transformation statistic z T is pretty good normal approximation even for small. Tables 3, 4 and 5 give the upper 10, 5 and 1% points of z MS, zms and z T, respectively. ote that the notation of z0.90, z0.95 and z0.99 mean the upper percent points of normal distribution. In Table 3, zms and z T are closer to the upper 10% point of normal distribution even when is small. In Table 4, the accuracy of approximation for z MS is good when is small. However the upper approximate percent points of z T are better as is large. Finally it may be seen from Table 5 that the values for z T is closer to the upper 1% point of normal distribution for any. Some histograms of the sample distributions for z MS, zms and z T by simulation are given in Figure 1 p = 10. Also, we compute the cases p = 3, 5, 7, and obtain results similar to these for the case p = 10. In conclusion, it is noted from various points of view that the normalizing transformation statistic improved for tha distribution function z T proposed in this paper is extremely good normal approximation and is useful for the multivariate normal test. 9 Conclusion and problems In this paper, we proposed a new definition for multivariate kurtosis. It is noticed that a new definition for multivariate kurtosis is based on fourth moment from definitions proposed by 16

Mardia 1970 and Srivastava 1984. Under normality, we derived the expectation, the variance and the third moment for a sample measure of new multivariate kurtosis. Further, standardized statistics and normalizing transformation statistic were given by using these results. Finally, we evaluated the accuracy of statistics derived in this paper by Monte Carlo simulation, and we recommend to use z T for the multivariate normality test. It is left as a future problem for the case when Σ is unknown. Acknowledgements The author s deepest appreciation goes to Dr. Kazuyuki Koizumi of Tokyo University of Science for giving many valuable advices. Also, special thanks to students in Seo Laboratory for their useful comments. References 1] Enomoto, R., Okamoto,. and Seo, T 010. On the distibution of test statistic using Srivastava s skewness and kurtosis, Technical Report o.10-07, Statistical Research Group, Hiroshima University. ] Jarque, C. M. and Bera, A. K. 1987. A test for normality of observations and regression residuals, International Statistical Review, 55, 163 17. 3] Koizumi, K., Okamoto,. and Seo, T 009. On Jarque-Bera tests for assessing multivariate normality, Journal of Statistics: Advances in Theory and Applications, 1, 07 0. 4] Konishi, S. 1981. ormalizing transformations of some statistics in multivariate analysis, Biometrika, 68, 647 651. 5] Looney, S. W. 1995. How to use test for univariate normality to assess multivariate normality, American Statician, 49, 64 70. 6] Malkovich, J. and Afifi, A. A. 1973. On tests for multivariate normality, Journal of American Statistical Association, 68, 176 179. 7] Mardia, K. V. 1970. Measures of multivariate skewness and kurtosis with applications, Biometrika, 57, 519 530. 8] Mardia, K. V. 1974. Applications of some measures of multivariate skewness and kurtosis in testing normality and robustness studies, Sankhyā B, 36, 115-18. 9] Mardia, K. V. and Foster, K. 1983. Omnibus tests of multinormality based on skewness and kurtosis, Communications in Statistics-Theory and Methods, 1, 07 1. 10] Mardia, K. V. and Kanazawa, M. 1983. The null distribution of multivariate kurtosis, Communications in Statistics-Simulation and Computation, 1, 569 576. 11] Okamoto,. and Seo, T. 010. On the distribution of multivariate sample skewness, Journal of Statistical Planning and Inference, 140, 809-816. 1] Royston, J. P. 1983. Some techniques for assessing multivariate normality based on the Shapiro-Wilk W, Applied Statistics, 3, 11 133. 17

13] Seo, T. and Ariga, M. 009 On the distribution of kurtosis test for multivariate normality, Submitted for publication. 14] Seo, T., Kanda, T. and Fujikoshi, Y. 1994. The effects on the distributions of sample canonical correlations under nonnormality, Communications in Statistics-Theory and Methods, 3, 615-68. 15] Shapiro, S. S. and Wilk, M. B. 1965. An analysis of variance test for normality complete samples, Biometrika, 5, 591 611. 16] Small,. J. H. 1980. Marginal skewness and kurtosis in testing multivariate normality, Applied Statistics,, 60 66. 17] Song, K. 001. Rènyi information, loglikelihood and an intrinsic distribution measure, Journal of Statistical Planning and Inference, 93, 51 69. 18] Srivastava, M. S. 1984. A measure of skewness and kurtosis and a graphical method for assessing multivariate normality, Statistics & Probability Letters,, 63 67. 19] Srivastava, M. S. and Hui, T. K. 1987. On assessing multivariate normality based on Shapiro-Wilk W statistic, Statistics & Probability Letters, 5, 15 18. 18

Table 1: Expectation and variance for z MS, zms and z T Expectation LT:0 Variance LT:1 z MS zms z T z MS zms z T p=3 0 0.135 0.001 0.109 0.840 0.991 0.631 50 0.086 0.001 0.036 0.936 1.000 0.794 100 0.061 0.000 0.014 0.970 1.00 0.878 00 0.043 0.001 0.006 0.987 1.003 0.931 400 0.09 0.001 0.003 0.99 1.000 0.96 800 0.0 0.000 0.000 0.998 1.00 0.983 p=5 0 0.133 0.000 0.109 0.848 1.00 0.63 50 0.085 0.000 0.036 0.938 1.00 0.79 100 0.06 0.001 0.013 0.967 0.999 0.878 00 0.044 0.001 0.005 0.984 1.000 0.931 400 0.031 0.001 0.001 0.990 0.998 0.96 800 0.0 0.000 0.000 0.995 0.999 0.979 p=7 0 0.13 0.00 0.111 0.851 1.005 0.63 50 0.085 0.001 0.036 0.939 1.00 0.795 100 0.06 0.001 0.013 0.965 0.997 0.876 00 0.044 0.001 0.004 0.981 0.997 0.99 400 0.09 0.001 0.003 0.994 1.00 0.964 800 0.0 0.001 0.000 0.995 0.999 0.980 p=10 0 1.133 0.000 0.109 0.849 1.003 0.63 50 1.085 0.001 0.036 0.940 1.003 0.795 100 0.060 0.001 0.014 0.970 1.00 0.877 00 0.044 0.001 0.005 0.983 0.999 0.930 400 0.03 0.000 0.001 0.988 0.996 0.960 800 0.01 0.000 0.001 0.996 1.000 0.980 19

Table : Skewness and kurtosis for z MS, zms and z T Skewness LT:0 Kurtosis LT:3 z MS zms z T z MS zms z T p=3 0.33.33 0.068 1.531 1.531.193 50 1.431 1.431 0.036 7.075 7.075.514 100 1.0 1.0 0.036 5.113 5.113.713 00 0.74 0.74 0.014 4.016 4.016.845 400 0.508 0.508 0.010 3.5 3.5.931 800 0.360 0.360 0.00 3.54 3.54.96 p=5 0.76.76 0.070 13.6 13.6.191 50 1.456 1.456 0.030 7.414 7.414.518 100 1.00 1.00 0.037 4.95 4.95.708 00 0.716 0.716 0.03 4.05 4.05.853 400 0.50 0.50 0.014 3.500 3.500.97 800 0.359 0.359 0.003 3.53 3.53.965 p=7 0.99.99 0.067 13.651 13.651.191 50 1.433 1.433 0.033 7.04 7.04.511 100 1.006 1.006 0.037 4.971 4.971.713 00 0.714 0.714 0.05 4.035 4.035.850 400 0.509 0.509 0.009 3.514 3.514.95 800 0.353 0.353 0.010 3.55 3.55.970 p=10 0 1.133 0.000 0.070 13.46 13.46.190 50 1.085 0.001 0.03 7.148 7.148.51 100 0.060 0.001 0.035 5.043 5.043.714 00 0.044 0.001 0.05 3.999 3.999.850 400 0.03 0.000 0.016 3.500 3.500.93 800 0.01 0.000 0.007 3.58 3.58.968 0

Table 3: The upper 10% point of z MS, zms and z T 0.90 z MS zms z T z0.90 p=3 0 0.99 1.3 1.1 1.8 50 1.160 1.87 1.16 1.8 100 1.5 1.307 1.33 1.8 00 1.64 1.318 1.54 1.8 400 1.77 1.313 1.64 1.8 800 1.86 1.310 1.73 1.8 p=5 0 0.993 1.5 1.13 1.8 50 1.163 1.90 1.17 1.8 100 1.8 1.310 1.35 1.8 00 1.61 1.315 1.5 1.8 400 1.75 1.311 1.6 1.8 800 1.83 1.307 1.71 1.8 p=7 0 0.996 1.8 1.14 1.8 50 1.161 1.89 1.17 1.8 100 1.1 1.303 1.31 1.8 00 1.57 1.310 1.48 1.8 400 1.8 1.318 1.68 1.8 800 1.80 1.305 1.69 1.8 p=10 0 0.993 1.4 1.13 1.8 50 1.16 1.90 1.17 1.8 100 1.6 1.308 1.34 1.8 00 1.57 1.311 1.49 1.8 400 1.69 1.305 1.57 1.8 800 1.84 1.309 1.7 1.8 1

Table 4: The upper 5% point of z MS, zms and z T 0.95 z MS zms z T z0.95 p=3 0 1.605 1.890 1.445 1.645 50 1.79 1.875 1.503 1.645 100 1.746 1.837 1.550 1.645 00 1.751 1.809 1.594 1.645 400 1.78 1.766 1.614 1.645 800 1.71 1.737 1.630 1.645 p=5 0 1.608 1.893 1.446 1.645 50 1.77 1.873 1.50 1.645 100 1.747 1.838 1.550 1.645 00 1.744 1.80 1.590 1.645 400 1.76 1.704 1.613 1.645 800 1.71 1.737 1.630 1.645 p=7 0 1.614 1.899 1.448 1.645 50 1.77 1.874 1.50 1.645 100 1.745 1.835 1.549 1.645 00 1.738 1.796 1.586 1.645 400 1.730 1.768 1.616 1.645 800 1.705 1.730 1.65 1.645 p=10 0 1.61 1.897 1.447 1.645 50 1.731 1.878 1.504 1.645 100 1.750 1.841 1.55 1.645 00 1.741 1.799 1.588 1.645 400 1.73 1.760 1.610 1.645 800 1.708 1.733 1.67 1.645

Table 5: The upper 1% point of z MS, zms and z T 0.99 z MS zms z T z0.99 p=3 0 3.161 3.581 1.719.36 50 3.066 3.57 1.937.36 100.95 3.035.090.36 00.797.864.04.36 400.664.706.6.36 800.574.601.98.36 p=5 0 3.0 3.65 1.7.36 50 3.074 3.65 1.939.36 100.917 3.05.087.36 00.773.848.197.36 400.665.693.53.36 800.567.594.93.36 p=7 0 3.168 3.610 1.71.36 50 3.079 3.70 1.940.36 100.917 3.07.087.36 00.773.839.19.36 400.665.706.6.36 800.559.596.87.36 p=10 0 3.03 3.66 1.7.36 50 3.076 3.67 1.939.36 100.939 3.049.095.36 00.77.838.19.36 400.654.695.55.36 800.561.588.89.36 3

4 0 4 0.7 0.7 0.7 0.6 0.6 0.6 0.5 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0.3 0. 0. 0. 0.1 0.1 0.1 0.0 0.0 0.0 4 0 4 4 0 4 z MS = 0 z MS = 0 z T = 0 0.7 0.7 0.7 0.6 0.6 0.6 0.5 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0.3 0. 0. 0. 0.1 0.1 0.1 0.0 0.0 0.0 4 0 4 4 0 4 4 0 4 z MS = 100 z MS = 100 z T = 100 0.7 0.7 0.7 0.6 0.6 0.6 0.5 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0.3 0. 0. 0. 0.1 0.1 0.1 0.0 0.0 0.0 4 0 4 4 0 4 4 0 4 z MS = 400 z MS = 400 z T = 400 Figure 1: The sample distributions for z MS, zms and z T for standard normal distribution p = 10. by simulation, and the density plot 4