Abstract
We propose a new omnibus test statistic for normality based on the Jarque–Bera test statistic. We give the exact first four moments of the null distribution for the statistic using a computer algebra system. Our proposed statistic is an improvement of Jarque–Bera test statistic. Then the cumulants of the standardized statistic satisfy the Cornish–Fisher assumption. We give a normalizing transformation of the statistic based on the Wilson–Hilferty transformation.
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References
Anscombe FJ, Glynn WJ (1983) Distribution of the kurtosis statistic b 2 for normal samples. Biometrika 70: 227–234
Barndorff-Nielsen OE, Cox DR (1989) Asymptotic techniques for use in statistics. Chapman and Hall, New York
Bowman KO, Shenton LR (1975) Omnibus test contours for departures from normality based on \({\surd b\sb{1}}\) and b 2. Biometrika 62: 243–250
D’Agostino R, Pearson ES (1973) Tests for departure from normality. Empirical results for the distributions of b 2 and \({\surd b\sb{1}}\) . Biometrika 60: 613–622
D’Agostino, R, Stephens, MA (eds) (1986) Goodness-of-fit techniques (Statistics, a series of textbooks and monographs). Marcel Dekker Inc, New York
Fisher RA (1930) The moments of the distribution for normal samples of measures of departure from normality. Proc R Soc Ser A 130: 16–28
Geary RC, Worlledge JPG (1947) On the computation of universal moments of tests of statistical normality derived from samples drawn at random from a normal universe. Application to the calculation of the seventh moment of b 2. Biometrika 34: 98–110
Hall P (1992) The bootstrap and Edgeworth expansion. Springer, New York
Hearn AC (2004) REDUCE User’s Manual, Ver. 3.8. Report
Hill GW, Davis AW (1968) Generalized asymptotic expansions of Cornish–Fisher type. Ann Math Stat 39: 1264–1273
Hsu CT, Lawley DN (1940) The derivation of the fifth and sixth moments of the distribution of b 2 in samples from a normal population. Biometrika 31: 238–248
Jarque CM, Bera AK (1987) A test for normality of observations and regression residuals. Internat Stat Rev 55: 163–172
Koizumi K, Okamoto N, Seo T (2009) On Jarque–Bera tests for assessing multivariate normality. J Stat Adv Theory Appl 1: 207–220
Niki N, Nakagawa S (1995) Computer algebra application to the distribution theory of multivariate statistics. In: Proceedings of the first Asian technological conference in mathematics, pp 689–696
Pearson ES, D’Agostino R, Bowman KO (1977) Tests for departure from normality: comparison of powers. Biometrika 64: 231–246
Poitras G (2006) More on the correct use of omnibus tests for normality. Econ Lett 90: 304–309
Shapiro SS, Wilk MB (1965) An analysis of variance test for normality: complete samples. Biometrika 52: 591–611
Sturt A, Ord JK (1994) Kendall’s advanced theory of statistics: vol 1 (6 edn): distribution theory. Arnold, London
Thadewald T, Buning H (2007) Jarque–Bera test and its competitors for testing normality—a power comparison. J Appl Stat 34: 87–105
Thode HC (2002) Testing for normality (Statistics, a series of textbooks and monographs). Marcel Dekker Inc, New York
Urzúa CM (1996) On the correct use of omnibus tests for normality. Econ Lett 53: 247–251
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This research is partially supported by the Japan Society for the Promotion of Science (JSPS), Grant–in–Aid for Scientific Research (C), No 22500266.
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Nakagawa, S., Hashiguchi, H. & Niki, N. Improved omnibus test statistic for normality. Comput Stat 27, 299–317 (2012). https://doi.org/10.1007/s00180-011-0258-0
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DOI: https://doi.org/10.1007/s00180-011-0258-0