The Annals of Statistics

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

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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


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