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An explicit Wishart moment formula for the product of two disjoint principal minors

Christian Genest, Frédéric Ouimet, Donald Richards

TL;DR

This work resolves the long-standing Wilks problem for the joint moments of two disjoint principal minors of a Wishart matrix by deriving an explicit formula that expresses the two-minor moment as the product of the corresponding marginal moments times a Gaussian hypergeometric function of matrix argument, {}_2F_1(-nu1,-nu2; alpha/2; P P^T). The method blends block-matrix analysis and Laplace-transform techniques with properties of hypergeometric functions of matrix arguments, yielding a general renormalized extension involving a matrix T and nu0. As a further contribution, the authors establish a stronger quantitative version of the Wishart analogue of the Gaussian product inequality for the case d=2, via a lower bound that involves {}_2F_1 and {}_0F_1. The results enhance the understanding of Wishart moments, connect to applications in multivariate statistics (e.g., regression and correlation analyses), and provide computational tools for evaluating matrix-argument hypergeometric functions in high dimensions.

Abstract

This paper provides the first explicit formula for the expectation of the product of two disjoint principal minors of a Wishart random matrix, solving a part of a broader problem put forth by Samuel S. Wilks in 1934 in the Annals of Mathematics. The proof makes crucial use of hypergeometric functions of matrix argument and their Laplace transforms. Additionally, a Wishart generalization of the Gaussian product inequality conjecture is formulated and a stronger quantitative version is proved to hold in the case of two minors.

An explicit Wishart moment formula for the product of two disjoint principal minors

TL;DR

This work resolves the long-standing Wilks problem for the joint moments of two disjoint principal minors of a Wishart matrix by deriving an explicit formula that expresses the two-minor moment as the product of the corresponding marginal moments times a Gaussian hypergeometric function of matrix argument, {}_2F_1(-nu1,-nu2; alpha/2; P P^T). The method blends block-matrix analysis and Laplace-transform techniques with properties of hypergeometric functions of matrix arguments, yielding a general renormalized extension involving a matrix T and nu0. As a further contribution, the authors establish a stronger quantitative version of the Wishart analogue of the Gaussian product inequality for the case d=2, via a lower bound that involves {}_2F_1 and {}_0F_1. The results enhance the understanding of Wishart moments, connect to applications in multivariate statistics (e.g., regression and correlation analyses), and provide computational tools for evaluating matrix-argument hypergeometric functions in high dimensions.

Abstract

This paper provides the first explicit formula for the expectation of the product of two disjoint principal minors of a Wishart random matrix, solving a part of a broader problem put forth by Samuel S. Wilks in 1934 in the Annals of Mathematics. The proof makes crucial use of hypergeometric functions of matrix argument and their Laplace transforms. Additionally, a Wishart generalization of the Gaussian product inequality conjecture is formulated and a stronger quantitative version is proved to hold in the case of two minors.
Paper Structure (6 sections, 4 theorems, 50 equations)

This paper contains 6 sections, 4 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. Results
  4. Proofs
  5. Numerical validation
  6. Technical lemmas

Key Result

Theorem 3.1

For any $d, p_1, p_2 \in \mathbb{N}$, let $p = p_1 + p_2$ with $p_1 \leq p_2$, and fix any degree-of-freedom $\alpha \in (p-1, \infty)$. Let $\Sigma\in \mathcal{S}_{++}^p$ be any $p \times p$ positive definite matrix expressed in the form $(\Sigma_{ij})$, where for each $i,j\in \{1,2\}$, the block $ Then, for every real $\nu_1 > -\alpha/2 + (p_1 - 1)/2$ and $\nu_2 > -\alpha/2 + (p_2 - 1)/2$, one h

Theorems & Definitions (9)

  • Conjecture 1.1
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Corollary 3.4
  • proof : Proof of Theorem \ref{['thm:1']}
  • proof : Proof of Corollary \ref{['cor:1']}
  • Lemma B.1
  • Lemma B.2