Abstract
The problem of testing for multivariate normality has received much attention. Among the myriad of tests available, we confine ourselves to three affine invariant and simple to implement tests. In order to compare the power of these tests against skew-normal distributions we use Monte Carlo simulations and isotones, a graphical device introduced by Mudholkar, Kollia, Lin and Patel (J. Roy. Statist. Soc. B 53, 1991, 221–232). To this end, we generalize the notion of a profile, a deterministic ideal sample used to construct isotones, to the bivariate case.
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References
Azzalini, A., 2005. The skew-normal distribution and related multivariate families. Scand. J. Statist. 32, 159–188.
Azzalini, A., Capitanio, A., 1998. Statistical applications of the multivariate skew-normal distribution. J. Roy. Statist. Soc. B 61, 579–602.
Baringhaus, L., Henze, N., 1992. Limit distributions for Mardia’s measure of multivariate skewness. Ann. Statist. 20, 1889–1902.
Eaton, M.R., Perlman, M.D., 1973. The non-singularity of generalized sample covariance matrices. Ann. Statist. 1, 710–717.
Gürtler, N., 2000. Asymptotic Results on the Class of BHEP tests for Multivariate Normality with Fixed and Variable Smoothing Parameter (in German). Doctoral Dissertation, University of Karlsruhe (TH), Germany.
Gupta, A.K., Chen, J.T., Tang, J., 2007. A multivariate two-factor skew model. Statistics 41, 301–307.
Henze, N., 1986. A probabilistic representation of the ‘skew-normal’ distribution. Scand. J. Statist. 13, 271–275.
Henze, N., Zirkler, B., 1990. A class of invariant consistent tests for multivariate normality. Commun. Statist. A 19, 3595–3617.
Henze, N. 1994. On Mardia’s kurtosis test for multivariate normality. Commun. Statist.-Theory Meth. 23, 1031–1045.
Henze, N., Wagner, Th., 1997. A new approach to the BHEP tests for multivariate normality. J. Multiv. Anal. 62, 1–23.
Henze, N., 2002. Invariant tests for multivariate normality: a critical review. Statist. Papers 43, 467–506.
Mardia, K.V., 1970. Measures of multivariate skewness and kurtosis with applications. Biometrika 57, 519–530.
Mudholkar, G.S., Kollia, G.D., Lin, C.T., Patel, K.R., 1991. A graphical procedure for comparing goodness-of-fit tests. J. R. Statist. Soc. B 53, 221–232.
Wilding, G.E., Mudholkar, G.S., 2007. Some modifications of the Z-tests of normality and their isotones. To appear in Statistical Methodology.
Wilding, G.E., Mudholkar, G.S., Kollia, G.D., 2007. Two sets of isotones for comparing tests of exponentiality. J. Statist. Plann. Infer. 137, 3815–3825.
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Ketterer, F., Klar, B. & Henze, N. The Use of Isotones for Comparing Tests of Normality Against Skew Normal Distributions. J Stat Theory Pract 3, 613–626 (2009). https://doi.org/10.1080/15598608.2009.10411950
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DOI: https://doi.org/10.1080/15598608.2009.10411950