## The Annals of Statistics

- Ann. Statist.
- Volume 8, Number 6 (1980), 1388-1390.

### Exact Noncentral Distributions of Wilks' $\Lambda$ and Wilks-Lawley $U$ Criteria as Mixtures of Incomplete Beta Functions: for Three Tests

G. William Walster and Marietta J. Tretter

#### Abstract

In this paper it is shown that in the general case the exact noncentral distributions of Wilks' $\Lambda$ and Wilks-Lawley $U$ can be obtained in a very straightforward manner. This completely eliminates the need for the more complicated inverse Mellon transform. It is first shown that any random variable whose moments satisfy Wilks' Type B integral equation (Type B random variables) has a distribution that can be represented as a mixture of incomplete beta functions. Then it is shown that the moments of Wilks' $\Lambda$ and Wilks-Lawley $U$ criteria can be written as mixtures of the moments of Type B random variables. Combining these results yields the noncentral distribution of Wilks' $\Lambda$ and Wilks-Lawley $U$ criteria as mixtures of incomplete beta functions for the following tests: equality of two dispersion matrices; MANOVA; and canonical correlation.

#### Article information

**Source**

Ann. Statist., Volume 8, Number 6 (1980), 1388-1390.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176345210

**Digital Object Identifier**

doi:10.1214/aos/1176345210

**Mathematical Reviews number (MathSciNet)**

MR594654

**Zentralblatt MATH identifier**

0459.62035

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62E15: Exact distribution theory

Secondary: 62H15: Hypothesis testing 62J10: Analysis of variance and covariance

**Keywords**

Incomplete beta function mixtures multivariate analysis of variance noncentral distributions

#### Citation

Walster, G. William; Tretter, Marietta J. Exact Noncentral Distributions of Wilks' $\Lambda$ and Wilks-Lawley $U$ Criteria as Mixtures of Incomplete Beta Functions: for Three Tests. Ann. Statist. 8 (1980), no. 6, 1388--1390. doi:10.1214/aos/1176345210. https://projecteuclid.org/euclid.aos/1176345210