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.
Publié le : 1980-11-14
Classification:
Incomplete beta function,
mixtures,
multivariate analysis of variance,
noncentral distributions,
62E15,
62H15,
62J10
@article{1176345210,
author = {Walster, G. William and Tretter, Marietta J.},
title = {Exact Noncentral Distributions of Wilks' $\Lambda$ and Wilks-Lawley $U$ Criteria as Mixtures of Incomplete Beta Functions: for Three Tests},
journal = {Ann. Statist.},
volume = {8},
number = {1},
year = {1980},
pages = { 1388-1390},
language = {en},
url = {http://dml.mathdoc.fr/item/1176345210}
}
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., Tome 8 (1980) no. 1, pp. 1388-1390. http://gdmltest.u-ga.fr/item/1176345210/