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.
Citation
G. William Walster. Marietta J. Tretter. "Exact Noncentral Distributions of Wilks' $\Lambda$ and Wilks-Lawley $U$ Criteria as Mixtures of Incomplete Beta Functions: for Three Tests." Ann. Statist. 8 (6) 1388 - 1390, November, 1980. https://doi.org/10.1214/aos/1176345210
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