Tests for Multivariate Normality based on the Empirical Moment Generating Function and other Criteria

Abstract

In this article, we present test statistics for assessing the compatibility of a multivariate random sample with the multivariate normal distribution. Three criteria are used to develop the proposed tests: the empirical multivariate moment generating function, mixed partial functional moments, and the empirical distribution function (EDF). The suggested tests are weighted integrals of the deviance or square deviance of the EMGF from the MGF, weighted integrals of the deviance or square deviance between the EMPM and the MPM, and EDF-type tests based on the stochastic ordering of the d-dimensional sample points. We derive computational forms of the proposed tests and conduct simulations to find their approximate critical values at a nominal level of 0.05. In addition, simulations approximate the powers of the proposed tests for various sample sizes when testing the bivariate normal distribution against a set of alternatives. The results show that the proposed tests compete well with some existing ones.