# Wishart exponential families on cones related to An graphs

## TL;DR

This paper investigates Wishart exponential families on cones associated with An graphs, providing explicit formulas for their key statistical functions and extending previous models to non-decomposable graphs.

## Contribution

It introduces a new approach to study Wishart NEF on cones related to An graphs, including non-decomposable cases, with explicit formulas for moments and covariance functions.

## Key findings

- Derived Riesz measures for Wishart NEF on cones QG and PG.
- Provided explicit formulas for mean, covariance, and higher moments.
- Extended Wishart models to non-decomposable An graphs.

## Abstract

Let G = An be the graph corresponding to the graphical model of nearest neighbour interaction in a Gaussian character. We study Natural Exponential Families( NEF) ofWishart distributions on convex cones QG and PG, where PG is the cone of positive definite real symmetric matrices with obligatory zeros prescribed by G, and QG is the dual cone of PG. The Wishart NEF that we construct include Wishart distributions considered earlier by Lauritzen (1996) and Letac and Massam (2007) for models based on decomposable graphs. Our approach is however different and allows us to study the basic objects ofWishart NEF on the cones QG and PG.We determine Riesz measures generating Wishart exponential families on QG and PG, and we give the quadratic construction of these Riesz measures and exponential families. The mean, inverse-mean, covariance and variance functions, as well as moments of higher order are studied and their explicit formulas are given.