On Wilks' joint moment formulas for embedded principal minors of Wishart random matrices

Author:

Genest C.1ORCID,Ouimet F.1ORCID,Richards D.2ORCID

Affiliation:

1. Department of Mathematics and Statistics McGill University Montréal Québec Canada

2. Department of Statistics Pennsylvania State University University Park Pennsylvania USA

Abstract

AbstractIn 1934, the American statistician Samuel S. Wilks derived remarkable formulas for the joint moments of embedded principal minors of sample covariance matrices in multivariate Gaussian populations, and he used them to compute the moments of sample statistics in various applications related to multivariate linear regression. These important but little‐known moment results were extended in 1963 by the Australian statistician A. Graham Constantine using Bartlett's decomposition. In this note, a new proof of Wilks' results is derived using the concept of iterated Schur complements, thereby bypassing Bartlett's decomposition. Furthermore, Wilks' open problem of evaluating joint moments of disjoint principal minors of Wishart random matrices is related to the Gaussian product inequality conjecture.

Funder

Natural Sciences and Engineering Research Council of Canada

Canada Research Chairs

Publisher

Wiley

Reference12 articles.

1. XX.—On the Theory of Statistical Regression

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4. Pfaffians, hafnians and products of real linear functionals

5. Genest C. Ouimet F. &Richards D.(2024).On the Gaussian product inequality conjecture for disjoint principal minors of Wishart random matrices 27pp.arXiv:2311.00202v2

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