## The Annals of Statistics

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

#### 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

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
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