Wishart distributions for decomposable covariance graph models

TL;DR

This paper introduces a new family of Wishart distributions tailored for decomposable covariance graph models, providing a rigorous analysis of their properties and expanding the tools available for statistical inference in this setting.

Contribution

It constructs covariance Wishart distributions for decomposable graphs and thoroughly studies their properties, filling a gap in the statistical modeling of covariance structures.

Findings

Defined covariance Wishart distributions on the cone P_G

Derived key properties of these distributions

Established their role analogous to classical Wishart in covariance graph models

Abstract

Gaussian covariance graph models encode marginal independence among the components of a multivariate random vector by means of a graph $G$. These models are distinctly different from the traditional concentration graph models (often also referred to as Gaussian graphical models or covariance selection models) since the zeros in the parameter are now reflected in the covariance matrix $\Sigma$, as compared to the concentration matrix $\Omega =\Sigma^{-1}$. The parameter space of interest for covariance graph models is the cone $P_G$ of positive definite matrices with fixed zeros corresponding to the missing edges of $G$. As in Letac and Massam [Ann. Statist. 35 (2007) 1278--1323], we consider the case where $G$ is decomposable. In this paper, we construct on the cone $P_G$ a family of Wishart distributions which serve a similar purpose in the covariance graph setting as those constructed by Letac and Massam [Ann. Statist. 35 (2007) 1278--1323] and Dawid and Lauritzen [Ann. Statist. 21 (1993) 1272--1317] do in the concentration graph setting. We proceed to undertake a rigorous study of these "covariance" Wishart distributions and derive several deep and useful properties of this class.