A test for the mean vector with fewer observations than the dimension under non-normality

In this article, we consider the problem of testing that the mean vector mu = 0 in the model x(j) = mu + C(zj), j = 1, ..., N, where z(j) are random p-vectors, z(j) = (z(ij), ..., z(pj))' and z(ij) are independently and identically distributed with finite four moments, i I..... P, i = 1,.... N; that is x(i) need not be normally distributed. We shall assume that C is a p x p non-singular matrix, and there are fewer observations than the dimension, N <= p. We consider the test statistic T = [N (x) over bar 'D(s)(-1 (x) over bar) - np/(n - 2)] / [2tr R(2) - p(2)/n](1/)2, where (x) over bar is the sample mean vector, S = (s(ij)) is the sample covariance matrix, D(S) diag (S(11), ..., S(pp)), R = D(s)(-1/2)SD(s)(-1/2) and n = N - 1. The asymptotic null and non-null distributions of the test statistic T are derived. (c) 2008 Published by Elsevier Inc.