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

2025-02-19

AI Summary

Exact hypothesis testing for random effects in two-factor and general factorial designs with multivariate normal data remains challenging, particularly in high dimensions. Method: We develop a novel approach grounded in the theory of noncentral Wishart distributions. Specifically, we establish the closure property of this distribution class by proving that mixtures—over common degrees of freedom—of noncentral Wishart distributions with respect to their noncentrality parameters remain noncentral Wishart. Leveraging this result, we extend the univariate χ²-type framework of Jones–Marchand to dimension *d*, deriving closed-form finite-sample distributions for random-effects test statistics. Contribution/Results: Our method overcomes the univariate restriction of Bilodeau’s work, enabling computationally tractable, asymptotically exact inference in *d*-dimensional settings. It provides the first rigorous matrix-variate distributional foundation for exact hypothesis testing in high-dimensional random-effects models, yielding practical, finite-sample inferential tools without reliance on asymptotic approximations.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

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.

Problem

• Extends noncentral Wishart mixture results to Wishart setting
• Derives test statistics for random effects in factorial designs
• Generalizes random effects testing to multidimensional normal data

Innovation

• Extends chi-square to Wishart setting
• Derives finite-sample distribution for test statistics
• Tests random effects in factorial designs