On Tests for Multivariate Normality

Abstract

Abstract The univariate skewness and kurtosis statistics, and b 2, and The W statistic proposed by Shapiro and Wilk are generalized to test a hypothesis of multivariate normality by use of S.N. Roy's union-intersection principle. These generalized statistics are invariant with respect to nonsingular matrix multiplication and vector addition. Two univariate test statistics, Kolmogorov-Smirnov and Cramér-Von Mises, are used to test whether transformed vector observations follow a χ2 distribution. The significance points, and powers against selected alternatives, of these five test statistics are obtained by Monte Carlo methods. These studies showed that adequate powers may be achieved for small sample sizes.