Graph presentations for moments of noncentral Wishart distributions and their applications

TL;DR

This paper derives formulas for moments of noncentral Wishart distributions using graph theory, providing explicit results and a combinatorial interpretation of Laguerre polynomial coefficients.

Contribution

It introduces graph-based formulas for moments of real and complex noncentral Wishart distributions, linking combinatorics with statistical moments.

Findings

Formulas for moments of real and complex noncentral Wishart distributions.

Explicit formulas for bivariate chi-square and 2x2 Wishart distributions.

Combinatorial interpretation of Laguerre polynomial coefficients.

Abstract

We provide formulas for the moments of the real and complex noncentral Wishart distributions of general degrees. The obtained formulas for the real and complex cases are described in terms of the undirected and directed graphs, respectively. By considering degenerate cases, we give explicit formulas for the moments of bivariate chi-square distributions and $2\times 2$ Wishart distributions by enumerating the graphs. Noting that the Laguerre polynomials can be considered to be moments of a noncentral chi-square distributions formally, we demonstrate a combinatorial interpretation of the coefficients of the Laguerre polynomials.