Riesz measures and Wishart laws associated to quadratic maps

Wishart laws and variance function on homogeneous cones

We present a systematic study of Riesz measures and their natural exponential families of Wishart laws on a homogeneous cone. We compute explicitly the inverse of the mean map and the variance function of a Wishart exponential family.

Wishart exponential families on cones related to An graphs

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 ofW…

Characterization of the Riesz Exponential Family on homogeneous cones

In the paper we present a characterization theorem of the Riesz measure and a Wishart exponential family on homogeneous cones through the invariance property of a natural exponential family under the action of the triangular group.

The randomization by Wishart laws and the Fisher information

Consider the centered Gaussian vector $X$ in $\R^n$ with covariance matrix $ Σ.$ Randomize $Σ$ such that $ Σ^{-1}$ has a Wishart distribution with shape parameter $p>(n-1)/2$ and mean $pσ.$ We compute the density $f_{p,σ}$ of $X$ as well as the Fisher information $I_p(σ)$ of the model $(f_{p,σ} )$ when $σ$ is the parameter. For using the Cramér-Rao inequality, we also compute the inverse of $I_p(σ)$. The important point of this note is the fact that this inverse is a linear combination of two simple operators on the space of symmetric matrices, namely $¶(σ)(s)=σs σ$ and $(σ\otimes σ)(s)=σ\, \m…

Gaussian和Laguerre（Wishart）族在软边缘处的极限定律的渐近展开

Asymptotic Expansions of the Limit Laws of Gaussian and Laguerre (Wishart) Ensembles at the Soft Edge

研究Gaussian和Laguerre族在软边缘处的极限定律的渐近展开，通过精确选择缩放常数和展开参数h，推导出极限定律的渐近展开式，并通过模拟数据验证结果。