## The Annals of Mathematical Statistics

### Sample Criteria for Testing Equality of Means, Equality of Variances, and Equality of Covariances in a Normal Multivariate Distribution

S. S. Wilks

#### Abstract

In this paper statistical test criteria are developed for testing equality of means, equality of variances and equality of covariances in a normal multivariate population of $k$ variables on the basis of a sample. More specifically, three statistical hypotheses are considered: (i) $H_{mvc}$, the hypothesis that the means are equal, the variances are equal, and covariances are equal, (ii) $H_{vc}$, the hypothesis that variances are equal and covariances are equal, irrespective of the values of the means, and (iii) $H_m$, the hypothesis of equal means, assuming variances are equal and covariances are equal. Test criteria $L_{mvc}$, $L_{vc}$, and $L_m$ are developed by the Neyman-Pearson method of likelihood ratios for testing $H_{mvc}$, $H_{vc}$ and $H_m$ respectively. The exact moments of each of the three test criteria when the three corresponding hypotheses are true are determined for any number $k$ of variables and for any size, $n$, of the sample for which the distributions exist. The exact distributions of $L_{mvc}$ and $L_{vc}$ are determined for $k = 2$ and $k = 3$, and the exact distribution of $L_m$ is found for any $k$; these are all beta (Pearson Type I) distributions. Tables of 5% and 1% points of $L_{mvc}$, $L_{vc}$ and $L_m$, based on Thompson's tables of percentage points of the Incomplete Beta Function, are given for certain values of $k$ and $n$ (Tables I and II). Also tables of values of approximate 5% and 1% points of $-n \ln L_{mvc}, -n \ln L_{vc}$ and $-n(k - 1) \ln L_m$ for large values of $n$ are given (Table III), based on the fact that these three quantities are approximately distributed according to chi-square laws for large values of $n$ with $\frac{1}{2}k(k + 3) - 3, \frac{1}{2}k(k + 1) - 2$, and $k - 1$ degrees of freedom respectively. A table (Table IV) is given which shows how accurate the resulting approximate 5% and 1% points of $L_{mvc}, L_c$ and $L_m$ are. The paper is written in two parts. In Part I the problem of testing the three hypotheses is discussed and the mathematical results are presented together with an illustrative example. Part II is given for the reader who wishes to study the mathematical derivation of the results.

#### Article information

Source
Ann. Math. Statist., Volume 17, Number 3 (1946), 257-281.

Dates
First available in Project Euclid: 28 April 2007

https://projecteuclid.org/euclid.aoms/1177730940

Digital Object Identifier
doi:10.1214/aoms/1177730940

Mathematical Reviews number (MathSciNet)
MR17498

Zentralblatt MATH identifier
0063.08259

JSTOR