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 μ=0 in the model xj=μ+Czj,j=1,…,N, where zj are random p-vectors, zj=(zij,…,zpj)′ and zij are independently and identically distributed with finite four moments, i=1,…,p,j=1,…,N; that is xi need not be normally distributed. We shall assume that C is a p×p non-singular matrix, and there are fewer observations than the dimension, N≤p. We consider the test statistic T=[Nx¯′Ds−1x¯−np/(n−2)]/[2trR2−p2/n]12, where x¯ is the sample mean vector, S=(sij) is the sample covariance matrix, DS= diag (s11,…,spp),R=Ds−12SDs−12 and n=N−1. The asymptotic null and non-null distributions of the test statistic T are derived.