In this article, we consider the problem of testing that the mean vector μ = 0 in the model xj = μ + C zj, j = 1, ..., N, where zj are random p-vectors, zj = (zi j, ..., zp j)′ and zi j 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 = [N over(x, -)′ Ds- 1 over(x, -) - n p / (n - 2)] / [2 tr R2 - p2 / n]frac(1, 2), where over(x, -) is the sample mean vector, S = (si j) is the sample covariance matrix, DS = diag (s11, ..., sp p), R = Ds- frac(1, 2) S Ds- frac(1, 2) and n = N - 1. The asymptotic null and non-null distributions of the test statistic T are derived. © 2008.
Srivastava, M. S. (2009). A test for the mean vector with fewer observations than the dimension under non-normality. Journal of Multivariate Analysis, 100(3), 518–532. https://doi.org/10.1016/j.jmva.2008.06.006