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 x j = μ + C z j , j = 1 , … , N , where z j are random p -vectors, z j = ( z i j , … , z p j ) ′ and z i j are independently and identically distributed with finite four moments, i = 1 , … , p , j = 1 , … , N ; that is x i 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 = [ N x ¯ ′ D s − 1 x ¯ − n p / ( n − 2 ) ] / [ 2 tr R 2 − p 2 / n ] 1 2 , where x ¯ is the sample mean vector, S = ( s i j ) is the sample covariance matrix, D S = diag ( s 11 , … , s p p ) , R = D s − 1 2 S D s − 1 2 and n = N − 1 . The asymptotic null and non-null distributions of the test statistic T are derived.