Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
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A Jarque-Bera type test for multivariate normality based on second-power skewness and kurtosis

  • Received : 2021.02.12
  • Accepted : 2021.05.17
  • Published : 2021.09.30
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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

Acknowledgement

This work was supported by 2021 Hongik University Research Fund.

References

  1. Anscombe FJ and Glynn WJ (1983). Distribution of the kurtosis statistic b2 for normal statistics, Biometrika, 70, 227-234. https://doi.org/10.1093/biomet/70.1.227
  2. D'Agostino RB (1970). Transformation to normality of the null distribution of g1, Biometrika, 57, 679-681. https://doi.org/10.1093/biomet/57.3.679
  3. D'Agostino RB and Stephens MA (1986). Goodness-of-Fit Techniques, Marcel Dekker, New York.
  4. 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. https://doi.org/10.1093/biomet/60.3.613
  5. 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.
  6. 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. https://doi.org/10.1080/02664763.2017.1415311
  7. De Wet T and Venter JH (1972). Asymptotic distributions of certain test criteria of normality, South African Statistical Journal, 6, 135-149.
  8. Doornik JA and Hansen H (2008). An omnibus test for univariate and multivariate normality, Oxford Bulletin of Economics and Statistics, 70, 927-939. https://doi.org/10.1111/j.1468-0084.2008.00537.x
  9. 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, https://doi.org/10.1080/03610918.2019.1661476
  10. 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. https://doi.org/10.1080/10629360600878449
  11. Fattorini L (1986). Remarks on the use of the Shapiro-Wilk statistic for testing multivariate normality, Statistica, 46, 209-217.
  12. 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. https://doi.org/10.1080/03610918.2017.1315771
  13. Henze N (2002) Invariant tests for multivariate normality: A critical review, Statistical Papers, 43, 467-506. https://doi.org/10.1007/s00362-002-0119-6
  14. Henze N, and Zirkler B (1990). A class of invariant consistent tests for multivariate normality, Communications in Statistics-Theory and Methods, 19, 3539-3617. https://doi.org/10.1080/03610929008830396
  15. 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. https://doi.org/10.1080/00949659208811407
  16. Jarque C and Bera A (1980). Efficient tests for normality, homoscedasticity and serial independence of regression residuals, Economics Letters, 6, 255-259. https://doi.org/10.1016/0165-1765(80)90024-5
  17. Jonsson K (2011). A robust test for multivariate normality, Economics Letters, 113, 199-201. https://doi.org/10.1016/j.econlet.2011.06.018
  18. Kim N (2004). An approximate Shapiro-Wilk statistic for testing multivariate normality, The Korean Journal of Applied Statistics, 17, 35-47. https://doi.org/10.5351/KJAS.2004.17.1.035
  19. Kim N (2005). The limit distribution of an invariant test statistic for multivariate normality, The Korean Communications in Statistics, 12, 71-86.
  20. Kim N (2015). Tests based on skewness and kurtosis for multivariate normality, Communications for Statistical Applications and Methods, 22, 361-375. https://doi.org/10.5351/CSAM.2015.22.4.361
  21. Kim N (2016). A robustified Jarque-Bera test for multivariate normality, Economics Letters, 140, 48-52. https://doi.org/10.1016/j.econlet.2016.01.007
  22. 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. https://doi.org/10.29220/CSAM.2020.27.5.501
  23. Kim N and Bickel PJ (2003). The limit distribution of a test statistic for bivariate normality, Statistica Sinica, 13, 327-349.
  24. Komunjer I (2007). Asymmetric power distribution: theory and applications to risk measurement, Journal of Applied Econometrics, 22, 891-921. https://doi.org/10.1002/jae.961
  25. Korkmaz S, Goksuluk D, and Zararsiz G (2014). MVN: An R package for assessing multivariate normality, The R Journal, 6, 151-162.
  26. Malkovich JF and Afifi AA (1973). On tests for multivariate normality, Journal of the American Statistical Association, 68, 176-179. https://doi.org/10.1080/01621459.1973.10481358
  27. Mardia KV (1970). Measures of multivariate skewness and kurtosis with applications, Biometrika, 57, 519-530. https://doi.org/10.1093/biomet/57.3.519
  28. Mardia KV (1974). Applications of some measures of multivariate skewness and kurtosis for testing normality and robustness studies, Sankhya A, 36, 115-128.
  29. Mardia KV (1975). Assessment of multinormality and the robustness of Hotelling's T2 test, Applied Statistics, 24, 163-171. https://doi.org/10.2307/2346563
  30. Mardia KV and Foster K (1983). Omnibus tests of multinormality based on skewness and kurtosis, Communications of Statistics-Theory and Methods, 12, 207-221. https://doi.org/10.1080/03610928308828452
  31. 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. https://doi.org/10.1080/0094965042000193233
  32. 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. https://doi.org/10.1080/03610929508831533
  33. Pearson ES (1956). Some aspects of the geometry of statistics, Journal of the Royal Statistical Society, Series A, 119, 125-146. https://doi.org/10.2307/2342880
  34. 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. https://doi.org/10.1017/S0305004100023987
  35. Rao CR (1948b). Test of significance in multivariate analysis, Biometrika, 35, 58-79. https://doi.org/10.1093/biomet/35.1-2.58
  36. Romeu JL and Ozturk A (1993). A comparative study of goodness-of-fit tests for multivariate normality, Journal of Multivariate Analysis, 46, 309-334. https://doi.org/10.1006/jmva.1993.1063
  37. Roy SN (1953). On a heuristic method of test construction and its use in multivariate analysis, Annals of Mathematical Statistics, 24, 220-238. https://doi.org/10.1214/aoms/1177729029
  38. Royston JP (1983). Some techniques for accessing multivariate normality based on the Shapiro-Wilk W, Applied Statistics, 32, 121-133. https://doi.org/10.2307/2347291
  39. Shapiro SS, and Francia RS (1972). An approximate analysis of variance test for normality, Journal of the American Statistical Association, 67, 215-216. https://doi.org/10.1080/01621459.1972.10481232
  40. Shapiro SS and Wilk MB (1965). An analysis of variance test for normality (complete samples), Biometrika, 52, 591-611. https://doi.org/10.1093/biomet/52.3-4.591
  41. Srivastava MS (1984). A measure of skewness and kurtosis and a graphical method for assessing multivariate normality, Statistics & Probability Letters, 2, 263-267. https://doi.org/10.1016/0167-7152(84)90062-2
  42. Srivastava MS and Hui TK (1987). On assessing multivariate normality based on Shapiro-Wilk W statistic, Statistics & Probability Letters, 5, 15-18. https://doi.org/10.1016/0167-7152(87)90019-8
  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. Thode Jr. HC (2002). Testing for Normality, Marcel Dekker, New York.
  45. 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. https://doi.org/10.1080/03610920802474465
  46. Zhou M and Shao Y (2014). A power test for multivariate normality, Journal of Applied Statistics, 41, 351-363. https://doi.org/10.1080/02664763.2013.839637