Browse > Article
http://dx.doi.org/10.29220/CSAM.2021.28.5.463

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

Kim, Namhyun (Department of Science, Hongik University)
Publication Information
Communications for Statistical Applications and Methods / v.28, no.5, 2021 , pp. 463-475 More about this Journal
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;
Citations & Related Records
Times Cited By KSCI : 2  (Citation Analysis)
연도 인용수 순위
1 Fattorini L (1986). Remarks on the use of the Shapiro-Wilk statistic for testing multivariate normality, Statistica, 46, 209-217.
2 Hanusz Z, Enomoto R, Seo T, and Koizumi K (2018). A Monte Carlo comparison of Jarque-Bera type tests and Henze-Zirkler test of multivariate normality, Communications in Statistics - Simulation and Computation, 47, 1439-1452.   DOI
3 Kim N (2016). A robustified Jarque-Bera test for multivariate normality, Economics Letters, 140, 48-52.   DOI
4 Mecklin CJ and Mundfrom DJ (2005). A Monte Carlo comparison of the Type I and Type II error rates of tests of multivariate normality, Journal of Statistical Computation and Simulation, 75, 93-107.   DOI
5 D'Agostino RB and Pearson ES (1973). Tests for departure from normality: Empirical results for the distributions of b2 and $\sqrt{b_1}$, Biometrika, 60, 613-622.   DOI
6 Anscombe FJ and Glynn WJ (1983). Distribution of the kurtosis statistic b2 for normal statistics, Biometrika, 70, 227-234.   DOI
7 D'Agostino RB (1970). Transformation to normality of the null distribution of g1, Biometrika, 57, 679-681.   DOI
8 D'Agostino RB and Stephens MA (1986). Goodness-of-Fit Techniques, Marcel Dekker, New York.
9 D'Agostino RB and Pearson ES (1974). Correction and amendment: Tests for departure from normality: empirical results for the distributions of b2 and $\sqrt{b_1}$, Biometrika, 61, 647.
10 Desgagne A and de Micheaux PL (2018). A powerful and interpretable alternative to the Jarque-Bera test of normality based on 2nd-power skewness and kurtosis, using the Rao's score test on the APD family, Journal of Applied Statistics, 45, 2307-2327.   DOI
11 Doornik JA and Hansen H (2008). An omnibus test for univariate and multivariate normality, Oxford Bulletin of Economics and Statistics, 70, 927-939.   DOI
12 Henze N (2002) Invariant tests for multivariate normality: A critical review, Statistical Papers, 43, 467-506.   DOI
13 Srivastava MS and Hui TK (1987). On assessing multivariate normality based on Shapiro-Wilk W statistic, Statistics & Probability Letters, 5, 15-18.   DOI
14 Thode Jr. HC (2002). Testing for Normality, Marcel Dekker, New York.
15 Villasenor-Alva, JA and Gonzalez-Estrada E (2009). A generalization of Shapiro-Wilk's test for multivariate normality, Communications in Statistics-Theory and Methods, 38, 1870-1883.   DOI
16 De Wet T and Venter JH (1972). Asymptotic distributions of certain test criteria of normality, South African Statistical Journal, 6, 135-149.
17 Farrell PJ, Salibian-Barrera M, and Naczk K (2007). On tests for multivariate normality and associated simulation studies, Journal of Statistical Computation and Simulation, 77, 1065-1080.   DOI
18 Henze N, and Zirkler B (1990). A class of invariant consistent tests for multivariate normality, Communications in Statistics-Theory and Methods, 19, 3539-3617.   DOI
19 Jarque C and Bera A (1980). Efficient tests for normality, homoscedasticity and serial independence of regression residuals, Economics Letters, 6, 255-259.   DOI
20 Kim N (2005). The limit distribution of an invariant test statistic for multivariate normality, The Korean Communications in Statistics, 12, 71-86.
21 Horswell RL and Looney SW (1992). A comparison of tests for multivariate normality that are based on measures of multivariate skewness and kurtosis, Journal of Statistical Computation and Simulation, 42, 21-38.   DOI
22 Jonsson K (2011). A robust test for multivariate normality, Economics Letters, 113, 199-201.   DOI
23 Kim N (2004). An approximate Shapiro-Wilk statistic for testing multivariate normality, The Korean Journal of Applied Statistics, 17, 35-47.   DOI
24 Kim N (2015). Tests based on skewness and kurtosis for multivariate normality, Communications for Statistical Applications and Methods, 22, 361-375.   DOI
25 Kim N and Bickel PJ (2003). The limit distribution of a test statistic for bivariate normality, Statistica Sinica, 13, 327-349.
26 Komunjer I (2007). Asymmetric power distribution: theory and applications to risk measurement, Journal of Applied Econometrics, 22, 891-921.   DOI
27 Mardia KV and Foster K (1983). Omnibus tests of multinormality based on skewness and kurtosis, Communications of Statistics-Theory and Methods, 12, 207-221.   DOI
28 Malkovich JF and Afifi AA (1973). On tests for multivariate normality, Journal of the American Statistical Association, 68, 176-179.   DOI
29 Mardia KV (1970). Measures of multivariate skewness and kurtosis with applications, Biometrika, 57, 519-530.   DOI
30 Mardia KV (1975). Assessment of multinormality and the robustness of Hotelling's T2 test, Applied Statistics, 24, 163-171.   DOI
31 Mudholkar GS, Srivastava DK, and Lin CT (1995). Some p-variate adaptations of the Shapiro-Wilk test of normality, Communications of Statistics-Theory and Methods, 24, 953-985.   DOI
32 Pearson ES (1956). Some aspects of the geometry of statistics, Journal of the Royal Statistical Society, Series A, 119, 125-146.   DOI
33 Rao CR (1948b). Test of significance in multivariate analysis, Biometrika, 35, 58-79.   DOI
34 Romeu JL and Ozturk A (1993). A comparative study of goodness-of-fit tests for multivariate normality, Journal of Multivariate Analysis, 46, 309-334.   DOI
35 Royston JP (1983). Some techniques for accessing multivariate normality based on the Shapiro-Wilk W, Applied Statistics, 32, 121-133.   DOI
36 Srivastava MS (1984). A measure of skewness and kurtosis and a graphical method for assessing multivariate normality, Statistics & Probability Letters, 2, 263-267.   DOI
37 Rao CR (1948a). Large sample tests of statistical hypotheses concerning several parameters with applications to problems of estimation, Mathematical Proceedings of the Cambridge Philosophical Society, 44, 50-57.   DOI
38 Kim N (2020). Omnibus tests for multivariate normality based on Mardia's skewness and kurtosis using normalizing transformation, Communications for Statistical Applications and Methods, 27, 501-510.   DOI
39 Korkmaz S, Goksuluk D, and Zararsiz G (2014). MVN: An R package for assessing multivariate normality, The R Journal, 6, 151-162.
40 Mardia KV (1974). Applications of some measures of multivariate skewness and kurtosis for testing normality and robustness studies, Sankhya A, 36, 115-128.
41 Roy SN (1953). On a heuristic method of test construction and its use in multivariate analysis, Annals of Mathematical Statistics, 24, 220-238.   DOI
42 Shapiro SS and Wilk MB (1965). An analysis of variance test for normality (complete samples), Biometrika, 52, 591-611.   DOI
43 Srivastava DK and Mudholkar GS (2003). Goodness of fit tests for univariate and multivariate normal models (In Khattree R and Rao CR eds.), Handbook of Statistics 22: Statistics in Industry, Elsevier, North Holland, 869-906.
44 Zhou M and Shao Y (2014). A power test for multivariate normality, Journal of Applied Statistics, 41, 351-363.   DOI
45 Enomoto R, Hanusz Z, Hara A, and Seo T (2019). Multivariate normality test using normalizing transformation for Mardia's multivariate kurtosis, Communications in Statistics-Simulation and Computation, 49, 684-698,   DOI
46 Shapiro SS, and Francia RS (1972). An approximate analysis of variance test for normality, Journal of the American Statistical Association, 67, 215-216.   DOI