Chi-Square Mixture Representations for the Distribution of the Scalar Schur Complement in a Noncentral Wishart Matrix

TL;DR

This paper demonstrates that the scalar Schur complement in a noncentral Wishart matrix can be expressed as a mixture of chi-square distributions, with specific mixture weights derived for rank-1 noncentrality cases.

Contribution

It introduces a novel mixture representation for the distribution of the scalar Schur complement in noncentral Wishart matrices, including explicit weights for rank-1 noncentrality.

Findings

Distribution is a mixture of central chi-square distributions.

Mixture weights for rank-1 case come from a noncentral beta mixture of Poisson distributions.

Provides a new analytical tool for understanding Wishart matrix distributions.

Abstract

We show that the distribution of the scalar Schur complement in a noncentral Wishart matrix is a mixture of central chi-square distributions with different degrees of freedom. For the case of a rank-1 noncentrality matrix, the weights of the mixture representation arise from a noncentral beta mixture of Poisson distributions.