ON JARQUE-BERA TESTS FOR ASSESSING MULTIVARIATE NORMALITY

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1 Journal of Statistics: Advances in Theory and Alications Volume, umber, 009, Pages 07-0 O JARQUE-BERA TESTS FOR ASSESSIG MULTIVARIATE ORMALITY KAZUYUKI KOIZUMI, AOYA OKAMOTO and TAKASHI SEO Deartment of Mathematical Information Science Faculty of Science Tokyo University of Science Jaan koizu70@yahoo.co.j Deartment of Food Sciences Faculty of Health and utrition Tokyo Seiei College Jaan Abstract In this aer, we consider some tests for the multivariate normality based on the samle measures of multivariate skewness and kurtosis. Samle measures of multivariate skewness and kurtosis were defined by Mardia [3], Srivastava [9] and so on. We derive new multivariate normality tests by using Mardia s and Srivastava s moments. For univariate case, Jarque and Bera [] roosed bivariate test using skewness and kurtosis. We roose some new bivariate tests for assessing multivariate normality which are natural extensions of Jarque-Bera test. Finally, the numerical results by Monte Carlo simulation are shown in order to evaluate accuracy of exectations, variances and uer ercentage oints for new test statistics roosed in this aer. 000 Mathematics Subject Classification: 6E0, 6H0. Keywords and hrases: Jarque-Bera test, multivariate skewness, multivariate kurtosis, normality test. Received June 3, Scientific Advances Publishers

2 08 KAZUYUKI KOIZUMI et al.. Introduction In statistical analysis, the test for normality is an imortant roblem. This roblem has been considered by many authors. Shairo and Wilk [8] s W-statistic is well known as the univariate normality test. For the multivariate case, some tests based on W-statistic were roosed by Malkovich and Afifi [], Royston [6], Srivastava and Hui [0] and so on. Mardia [3] and Srivastava [9] gave different definitions of the multivariate measures of skewness and kurtosis, and discussed the test statistics using these measures for assessing multivariate normality, resectively. Mardia [4] derived exact exectations and variances of multivariate samle skewness and kurtosis, and discussed their asymtotic distributions. Srivastava [9] s samle measures of multivariate skewness and kurtosis have been discussed by many authors. Seo and Ariga [7] derived normalizing transformations of test statistic using Srivastava s kurtosis by the asymtotic exansion. Okamoto and Seo [5] derived the exact exectation and variance of Srivastava s skewness and imroved test statistic for assessing multivariate normality. On the other hand, for univariate samle case, Jarque and Bera [] roosed the bivariate test using skewness and kurtosis. Imroved Jarque-Bera tests have been considered by many authors (see, e.g., Urzúa []). It seems that Jarque-Bera test for multivariate case has not been considered by any author. Our urose is to roose new Jarque- Bera tests for assessing multivariate normality by using Mardia s and Srivastava s skewness and kurtosis, resectively. ew test statistics are asymtotically distributed as χ - distribution. We investigate accuracy of exectations, variances and uer ercentage oints for multivariate Jarque-Bera tests by Monte Carlo simulation.. Multivariate Measures of Skewness and Kurtosis.. Mardia s skewness and kurtosis Let x ( x, x, x ), and y ( y, y, y ), be random - = K = K vectors distributed identically and indeendently with mean vector µ =

3 O JARQUE-BERA TESTS FOR ASSESSIG 09 ( µ µ µ ),, K, and covariance matrix. Mardia [3] has defined the oulation measures of multivariate skewness and kurtosis as 3 [{( µ ) = E x ( y )} ], βm, µ [{( µ ) = E x ( x )} ], βm, µ resectively. When =, βm, and β M, are reduced to the ordinary univariate measures. It is obvious that for any symmetric distribution about µ β M, = 0. For a normal distribution ( µ, ),, βm, = 0, βm, = ( + ). To give the samle counterarts of β M, and β M,, let x, x, K, x be samles of size from a multivariate -dimensional oulation. And let x and S be the samle mean vector and the samle covariance matrix as follows: x = j, x j= S = ( )( ) x j x x j x, j= resectively. Then Mardia [3] has defined the samle measures of skewness and kurtosis by resectively. 3 b M, = x i j i= j= b {( x x ) S ( x )}, M, = i i x i= {( x x ) S ( x )},

4 0 KAZUYUKI KOIZUMI et al. Mardia [3, 4] has given the following lemma. Lemma (Mardia [3, 4]). The exact exectation of b M, and exectation and variance of b M, when the oulation is ( µ, ) are given by ( + ) ( b, ) = {( + ) ( ) }, ( )( 3) E M ( b ) = E M, ( + )( ), + 8( + )( 3) Var( bm, ) = ( )( ). ( ) ( )( ) Furthermore, Mardia [3] has obtained asymtotic distributions of b M, and M, b and used them to test the multivariate normality. Theorem (Mardia [3]). Let b M, and b M, be the samle measures of multivariate skewness and kurtosis, resectively, on the basis of a random samle of size drawn from ( µ, ), > 0. Then z M, b, 6 M is asymtotically distributed as χ - distribution with f ( + ) ( + ) 6 degrees of freedom, and ( ) ( bm, 8 + ( )) zm, + is asymtotically distributed as ( 0, ). By making reference to moments of b M, and b M,, Mardia [4] considered the following aroximate test statistics as cometitors of z and z : M, M, ( + ) ( + ) ( + 3) {( + ) ( + ) 6} zm, = bm, ~ 6 χ f (.)

5 asymtotically, and O JARQUE-BERA TESTS FOR ASSESSIG ( + 3)( + 5){( + ) bm, ( + ) ( )} ~ 8( + )( 3)( )( + ) zm, = ( 0, ) (.) asymtotically. It is noted that M,, f z is formed so that E( z M ) =... Srivastava s skewness and kurtosis Let Γ = ( γ, γ, K, γ ) be an orthogonal matrix such that Γ Γ = D λ, where Dλ = diag( λ, λ, K, λ ). ote that λ, λ, K, λ are the eigenvalues of. Then, Srivastava [9] defined the oulation measures of multivariate skewness and kurtosis as follows: β β S, S, = = i= i= 3 E[( vi θi ) ], 3 λ E[( vi θi ) ], λ i i 4 resectively, where v i = γ i x and θ i = γ iµ ( i =,, K, ). We note that βs, = 0, βs, = 3 under a multivariate normal oulation. Let H = ( h, h, K, h ) be an orthogonal matrix such that H SH = D w, where D w = diag( w, w, K, w ) and w, w, K, w are the eigenvalues of S. Then, Srivastava [9] defined the samle measures of multivariate skewness and kurtosis as follows: b S, 3 3 = ( ), wi vij vi i= j= b 4 = wi ( vij v ), S, i i= j= resectively, where vij = h i x j, vi = ( ) v. = ij j

6 KAZUYUKI KOIZUMI et al. Srivastava [9] obtained the following lemma: Lemma (Srivastava [9]). For large, the exectations of b S, and b S, and exectation and variance of S, ( µ, ) are given by 6 E( bs, ) = 0, E( bs, ) =, 4 E( bs, ) = 3, Var( bs, ) =. b when the oulation is By using Lemma, Srivastava [9] derived the following theorem: Theorem (Srivastava [9]). Let b S, and b S, be the samle measures of multivariate skewness and kurtosis, resectively, on the basis of a random samle of size drawn from ( µ, ), > 0. Then zs, b, 6 S is asymtotically distributed as χ - distribution with degrees of freedom, and ( bs, 3) 4 z S, is asymtotically distributed as ( 0, ). Further, Okamoto and Seo [5] gave the exectation of multivariate samle skewness b S, without using Taylor exansion. By using the same way as Okamoto and Seo [5], we can obtain the exectation and variance of multivariate samle kurtosis b S,. Hence, we can get the following lemma: of S, Lemma 3. For large, we give mean of b S, and mean and variance b when the oulation is ( µ, ).

7 O JARQUE-BERA TESTS FOR ASSESSIG 3 ( b ) = E S, 6( ) ( + )( + 3), ( b ) = E S, Var( b S ) =, 3( ), + 4 ( ) ( 3) ( ) ( )( ), resectively. Also, Seo and Ariga [7] gave asymtotic exansions of exectation and variance of b S,. By making reference to moments of b S, and b S,, we consider following aroximate test statistics as cometitors of z S, and z S, : asymtotically, and ( + )( + 3) 6( ) zs, = b S, ~ χ (.3) ( + 3)( + 5){( + ) bs, 3( )} ~ 4( )( 3) zs, = ( 0, ) (.4) asymtotically. 3. Multivariate Jarque-Bera Tests In this section, we consider new tests for multivariate normality when the oulation is ( µ, ). From Theorem, we roose a new test statistic using Mardia s measures as follows: MJBM = b M, 6 + ( b, ( )) M +. 8( + ) MJB M statistic is asymtotically distributed as χ f + - distribution. From Theorem, we roose a new test statistic using Srivastava s measures as follows:

8 4 KAZUYUKI KOIZUMI et al. MJB S = b S, 6 + ( b, 3) S. 4 MJB statistic is asymtotically distributed as - distribution. S Further, by using (.) and (.), modified χ + MJB M is given by MJB M = zm, + zm,. In the same as MJB M, this statistic distribution asymtotically. M χ f + MJB is distributed as - Also, by using (.3) and (.4), modified MJB S is given by MJB S = zs, + zs,. In the same as MJB S, this statistic distribution asymtotically. S χ + MJB is distributed as - 4. Simulation Studies Accuracy of exectations, variances and uer ercentage oints of multivariate Jarque-Bera tests MJB M MJBS, MJBM, and MJB S is evaluated by Monte Carlo simulation study. Simulation arameters are as follows: = 3, 0, 0, = 0, 50, 00, 00, 400, 800. As a numerical exeriment, we carry out 00,000 and,000,000 relications for the case of Mardia s measures and Srivastava s measures, resectively. and From Tables -, exectations of aroximate χ statistic MJB S are invariant for any samle sizes. That is, MJB M MJB M and MJB S are almost close to exact exectations even for small. However, accuracy of exectation of MJB M and MJB S is not good esecially for small. We note that exectations of MJB M and MJB S are convergence in those of exact χ - distribution for large. Hence, it may

9 be notice that both MJB S, resectively. O JARQUE-BERA TESTS FOR ASSESSIG 5 MJB M and On the other hand, from Tables -, variances of are larger than those of M MJB S are imrovements of MJB M and MJB M and MJB S MJB and MJB. The tendency aears well when samle size is small. But the coming off values of S MJB M and MJB S are more than those of MJB M and MJB S. Therefore, there is a tendency for magnitude of variance to become large. Finally, in Table 3, we give uer ercentage oints of MJB M and MJB M, by using Mardia s skewness and kurtosis. MJB M tends to be conservative. But MJB M is closer to the uer ercentage oints of χ f + - distribution even when the samle size is small. In Table 4, we give uer ercentage oints of MJB S and MJB S by using Srivastava s skewness and kurtosis. We note that the tendency is similar to the case using Mardia s moments. 5. Concluding Remarks For univariate case, Jarque-Bera test is well known to mount easily on ractical use. In this aer, we have roosed four new test statistics for assessing multivariate normality. We recommend to use MJB M and MJB S from the view of ractical use in the case of multivariate normality tests. But aroximations of exectations, variances and uer ercentage oints of MJB M and MJB S are not good when the samle size is small. Also, in this aer, we roose imroved multivariate M normality tests MJB and MJB S. Hence, we have been imroved exectations and uer ercentage oints of MJB M and MJB S. But variances of MJB M and MJB S are not imroved. This is a future roblem. We recommend to use MJB M and MJB S from the asect of

10 6 KAZUYUKI KOIZUMI et al. accuracy of aroximations to uer ercentage oints of test statistics esecially for small. References [] C. M. Jarque and A. K. Bera, A test for normality of observations and regression residuals, Int. Statist. Rev. 55 (987), [] J. R. Malkovich and A. A. Afifi, On tests for multivariate normality, J. Amer. Statist. Assoc. 68 (973), [3] K. V. Mardia, Measures of multivariate skewness and kurtosis with alications, Biometrika 57 (970), [4] K. V. Mardia, Alications of some measures of multivariate skewness and kurtosis in testing normality and robustness studies, Sankhyá B 36 (974), 5-8. [5]. Okamoto and T. Seo, On the Distribution of Multivariate Samle Skewness for Assessing Multivariate ormality, Technical Reort o. 08-0, Statistical Research Grou, Hiroshima University, (008). [6] J. P. Royston, Some techniques for assessing multivariate normality based on the Shairo-Wilk W, Al. Statist. 3 (983), -33. [7] T. Seo and M. Ariga, On the Distribution of Kurtosis Test for Multivariate ormality, Technical Reort o , Statistical Research Grou, Hiroshima University, (006). [8] S. S. Shairo and M. B. Wilk, An analysis of variance test for normality (comlete samles), Biometrika 5 (965), [9] M. S. Srivastava, A measure of skewness and kurtosis and a grahical method for assessing multivariate normality, Statist. & Prob. Lett. (984), [0] M. S. Srivastava and T. K. Hui, On assessing multivariate normality based on Shairo-Wilk W statistic, Statist. & Prob. Lett. 5 (987), 5-8. [] C. M. Urzúa, On the correct use of omnibus tests for normality, Econom. Lett. 90 (996),

11 O JARQUE-BERA TESTS FOR ASSESSIG 7 Table. Exectations and variances of MJB M and MJB M E ( MJB M ) E( MJB M ) f + Var ( MJB M ) Var ( MJB M ) ( f + )

12 8 KAZUYUKI KOIZUMI et al. Table. Exectations and variances of MJB S and MJB S E ( MJB S ) E( MJB S ) + Var ( MJB S ) Var ( MJB S ) ( + )

13 O JARQUE-BERA TESTS FOR ASSESSIG 9 Table 3. The uer ercentage oints of MJB M and MJB M MJB M MJB M χ f ( 0.05)

14 0 KAZUYUKI KOIZUMI et al. Table 4. The uer ercentage oints of MJB S and MJB S MJB S MJB S χ + ( 0.05)