Some Recent Results on Asymptotic Expansions of Multivariate Test Statistics for Mean Vectors Under Nonnormality

In this paper we review some recent results on asymptotic expansions of some multivariate test statistics for mean vectors under nonnormality. It is assumed that the samples are taken from a general distribution satisfying Cramér's condition. Some methods are discussed, through the derivation of asymptotic expansions of the distributions of Hotelling's T²-statistic, one-way MANOVA test statistics, and some test statistics for a multivariate linear hypothesis on regression coefficients. The results are restricted to the null distributions, but we shall see that the methods could be extended to the non-null case. Relating to the methods, asymptotic expansions are given for the distributions of the sample mean vector, the sample covariance matrix, and multivariate t-statistics. We also give some results on robustness of the test statistics for departures from normality.