On the Distribution of the Sphericity Test Criterion in Classical and Complex Normal Populations Having Unknown Covariance Matrices

Abstract

Let $\mathbf{x}: p \times 1$ be distributed $N(\mathbf{\mu}, \mathbf{\Sigma})$ where $\mathbf{\mu}$ and $\mathbf{\Sigma}$ are both unknown. Let $\mathbf{S}$ be the sum of product matrix of a sample of size $N$. To test the hypothesis of sphericity, namely, $H_0:\mathbf{\Sigma} = \sigma^2\mathbf{I}_p$, where $\sigma^2 > 0$ is unknown, against $H_1:\mathbf{\Sigma} \neq \sigma^2\mathbf{I}_p$, Mauchly [10] obtained the likelihood ratio test criterion for $H_0$ in the form $W = |\mathbf{S}|/\lbrack(\operatorname{tr} \mathbf{S})/p\rbrack^p$. Thus the criterion $W$ is a power of the ratio of the geometric mean and the arithmetic mean of the roots $\theta_1, \theta_2, \cdots, \theta_p$ of $|\mathbf{S} - \theta\mathbf{I}| = \mathbf{0}$ (see Anderson [1]). In the null case, Machly [10] gave the density of $W$ for $p = 2$ and Consul [3], [4] for any $p$ in terms of Meijer's $G$-function defined in the next section. In this paper we have obtained the general moments of $W$ both in real and complex cases for arbitrary covariance matrices, and also the corresponding distributions of $W$ in terms of the $G$-function.