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

Abstract

In this article, we consider the problem of testing that the mean vector $\mu =0$ in the model $x_j=\mu +C{z}_j,j=1,\dots ,N$, where $z_j$ are random $p$-vectors, $z_j={(z_{ij},\dots ,z_{pj})}^{\prime }$ and $z_{ij}$ are independently and identically distributed with finite four moments, $i=1,\dots ,p,j=1,\dots ,N$; that is $x_i$ need not be normally distributed. We shall assume that $C$ is a $p\times p$ non-singular matrix, and there are fewer observations than the dimension, $N\le p$. We consider the test statistic

$$T=[N{\overline{x}}^{\prime }{D}_s^{-1}\overline{x}-np/(n-2)]/{[2\mathrm{tr}{R}^2-p^2/n]}^{\frac{1}{2}}\text{,}$$

where $\overline{x}$ is the sample mean vector, $S=\left(s_{ij}\right)$ is the sample covariance matrix, $D_S=$ diag $(s_{11},\dots ,s_{pp}),R=D_s^{-\frac{1}{2}}S{D}_s^{-\frac{1}{2}}$ and $n=N-1$. The asymptotic null and non-null distributions of the test statistic $T$ are derived.