On tests for multivariate normality and associated simulation studies

We study the empirical size and power of some recently proposed tests for multivariate normality (MVN) and compare them with the existing proposals that performed best in previously published studies. We show that the Royston's [Royston, J.P., 1983b, Some techniques for assessing multivariate normality based on the Shapiro-Wilk W. Applied Statistics, 32,121-133.] extension to the Shapiro and Wilk [Shapiro, S.S. and Wilk, M.B., 1965, An analysis of variance test for normality (complete samples). Biometrika, 52, 591-611.] test is unable to achieve the nominal significance level, and consider a subsequent extension proposed by Royston [Royston, J.P., 1992, Approximating the Shapiro-Wilk W-Test for non-normality. Statistics and Computing, 2, 117-119.] to correct this problem, which earlier studies appear to have ignored. A consistent and invariant test proposed by Henze and Zirkler [Henze, N. and Zirkler, B., 1990, A class of invariant consistent tests for multivariate normality. Communications in Statistics- Theory and Methods, 19, 3595-3617.] is found to have good power properties, particularly for sample sizes of 75 or more, while an approach suggested by Royston [Royston, J.P, 1992, Approximating the Shapiro-Wilk W-Test for non-normality. Statistics and Computing, 2, 117-119.] performs effectively at detecting departures from MVN for smaller sample sizes. We also compare our results to those of previous simulation studies, and discuss the challenges associated with generating multivariate data for such investigations.