Moments of minors of Wishart matrices

TL;DR

This paper derives formulas for the expectation and covariance of minors of Wishart matrices, providing insights into their moments which are relevant for multivariate statistical analysis.

Contribution

It introduces new formulas for the moments of minors of Wishart matrices, advancing understanding of their statistical properties.

Findings

Explicit formulas for the expectation of minors

Covariance matrix of minors derived

Application to moments of sample covariance matrices

Abstract

For a random matrix following a Wishart distribution, we derive formulas for the expectation and the covariance matrix of compound matrices. The compound matrix of order $m$ is populated by all $m\times m$-minors of the Wishart matrix. Our results yield first and second moments of the minors of the sample covariance matrix for multivariate normal observations. This work is motivated by the fact that such minors arise in the expression of constraints on the covariance matrix in many classical multivariate problems.