The Annals of Mathematical Statistics

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

K. C. S. Pillai and B. N. Nagarsenker

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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.

Article information

Source
Ann. Math. Statist., Volume 42, Number 2 (1971), 764-767.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177693427

Digital Object Identifier
doi:10.1214/aoms/1177693427

Zentralblatt MATH identifier
0218.62051

JSTOR
links.jstor.org

Citation

Pillai, K. C. S.; Nagarsenker, B. N. On the Distribution of the Sphericity Test Criterion in Classical and Complex Normal Populations Having Unknown Covariance Matrices. Ann. Math. Statist. 42 (1971), no. 2, 764--767. doi:10.1214/aoms/1177693427. https://projecteuclid.org/euclid.aoms/1177693427


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