Hotelling's one-sample and two-sample [formula omitted] tests and the multivariate Behrens–Fisher problem under nonnormality

Abstract

This paper provides a powerful method for obtaining an asymptotic expansion for the expectation of a function of the normalized sample mean vector N 1 / 2 U ¯ and the sample covariance matrix S U under general distributions, without using the Edgeworth expansion of N - 1 / 2 ∑ i = 1 N U i and N - 1 / 2 ∑ i = 1 N vech ( U i U i ′ - Σ ) . It is shown that asymptotic expansions of the nonnull distributions of some multivariate test statistics on mean vectors under general distributions are derived in a unified way, by finding the differential operator associated with the expectation according to situations under consideration. Unlike Kano [1995. An asymptotic expansion of the distribution of Hotelling's T 2 -statistic under general distributions. Amer. J. Math. Manage. Sci. 15, 317–341] and Fujikoshi [1997. An asymptotic expansion for the distribution of Hotelling's T 2 -statistic under nonnormality. J. Multivariate Anal. 61, 187–193], our routine for getting asymptotic expansions is to collect some patterned derivatives, without constructing the Edgeworth expansion of some basic statistics. In this sense, our approach seems to be more convenient, at least, for Hotelling's T 2 -type statistics and other related statistics on mean vectors.