On Mardia’s Tests of Multinormality

Abstract

Classical multivariate analysis is based on the assumption that the data come from a multivariate normal distribution. The tests of multinor- mality have therefore received very much attention. Several tests for assess- ing multinormality, among them Mardia’s popular multivariate skewness and kurtosis statistics, are based on standardized third and fourth moments. In Mardia’s construction of the affine invariant test statistics, the data vectors are first standardized using the sample mean vector and the sample covariance matrix. In this paper we investigate whether, in the test construction, it is ad- vantagoeus to replace the regular sample mean vector and sample covariance matrix by their affine equivariant robust competitors. Limiting distributions of the standardized third and fourth moments and the resulting test statistics are derived under the null hypothesis and are shown to be strongly dependent on the choice of the location vector and scatter matrix estimate. Finally, the effects of the modification on the limiting and finite-sample efficiencies are illustrated by simple examples in the case of testing for the bivariate normal- ity. In the cases studied, the modifications seem to increase the power of the tests.