On noncentral Wishart mixtures of noncentral Wisharts and their use for testing random effects in factorial design models

TL;DR

This paper extends the understanding of noncentral Wishart distributions by showing that mixtures of such distributions with the same degrees of freedom also form a noncentral Wishart distribution, and applies this to test random effects in factorial designs.

Contribution

It generalizes a key result from chi-square to Wishart distributions and derives finite-sample distributions for random effects tests in multivariate factorial models.

Findings

Noncentral Wishart mixture yields a noncentral Wishart distribution.

Finite-sample distribution of test statistics for random effects derived.

Applicable to general factorial design models.

Abstract

It is shown that a noncentral Wishart mixture of noncentral Wishart distributions with the same degrees of freedom yields a noncentral Wishart distribution, thereby extending the main result of Jones and Marchand [Stat 10 (2021), Paper No. e398, 7 pp.] from the chi-square to the Wishart setting. To illustrate its use, this fact is then employed to derive the finite-sample distribution of test statistics for random effects in a two-factor factorial design model with $d$-dimensional normal data, thereby broadening the findings of Bilodeau [ArXiv (2022), 6 pp.], who treated the case $d = 1$. The same approach makes it possible to test random effects in more general factorial design models.