Wishart exponential families on cones related to An graphs

正定矩阵锥上强混合时间序列的Wishart核密度估计

Wishart kernel density estimation for strongly mixing time series on the cone of positive definite matrices

提出Wishart核密度估计器用于正定矩阵锥上的密度估计，该估计器具有边界感知性，能缓解边界偏差，并在混合条件下建立了均方误差、一致强相合性和渐近正态性，模拟和实例表明其优于其他方法。

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.

基于Wishart核密度估计器的对称正定矩阵分布两样本检验

A two-sample test for symmetric positive definite matrix distributions using Wishart kernel density estimators

本文提出一种非参数两样本检验方法，利用Wishart核密度估计器检测对称正定矩阵分布的差异，通过闭式叠加避免数值积分，提供了一种核方法替代Lukić的经验拉普拉斯变换检验。

Wishart分布的Stein方法

Stein's method for the Wishart distribution

本文针对正定矩阵锥上的Wishart分布发展了Stein方法，建立了基于扩展生成元的Stein刻画、转移半群、Stein方程的解及其正则性估计，并通过四个应用展示了新方法。

Wishart distributions for decomposable covariance graph models

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 $Σ$, as compared to the concentration matrix $Ω=Σ^{-1}$. 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 ed…

Hoffmann-Jørgensen Inequalities for Random Walks on the Cone of Positive Definite Matrices

We consider random walks on the cone of $m \times m$ positive definite matrices, where the underlying random matrices have orthogonally invariant distributions on the cone and the Riemannian metric is the measure of distance on the cone. By applying results of Khare and Rajaratnam (Ann. Probab., 45 (2017), 4101--4111), we obtain inequalities of Hoffmann-Jørgensen type for such random walks on the cone. In the case of the Wishart distribution $W_m(a,I_m)$, with index parameter $a$ and matrix parameter $I_m$, the identity matrix, we derive explicit and computable bounds for each term appearing i…

A Class of Infinitely Divisible Multivariate and Matrix Gamma Distributions and Cone-valued Generalised Gamma Convolutions

Classes of multivariate and cone valued infinitely divisible Gamma distributions are introduced. Particular emphasis is put on the cone-valued case, due to the relevance of infinitely divisible distributions on the positive semi-definite matrices in applications. The cone-valued class of generalised Gamma convolutions is studied. In particular, a characterisation in terms of an Itô-Wiener integral with respect to an infinitely divisible random measure associated to the jumps of a Lévy process is established. A new example of an infinitely divisible positive definite Gamma random matrix is intr…

Integral Transform Methods in Goodness-of-Fit Testing, II: The Wishart Distributions

We initiate the study of goodness-of-fit testing when the data consist of positive definite matrices. Motivated by the recent appearance of the cone of positive definite matrices in numerous areas of applied research, including diffusion tensor imaging, models of the volatility of financial time series, wireless communication systems, and the analysis of polarimetric radar images, we apply the method of Hankel transforms of matrix argument to develop goodness-of-fit tests for Wishart distributions with given shape parameter and unknown scale matrix. We obtain the limiting null distribution of …

A method to construct exponential families by representation theory

In this paper, we give a method to construct "good" exponential families systematically by representation theory. More precisely, we consider a homogeneous space $G/H$ as a sample space and construct an exponential family invariant under the transformation group $G$ by using a representation of $G$. The method generates widely used exponential families such as normal, gamma, Bernoulli, categorical, Wishart, von Mises, Fisher-Bingham and hyperboloid distributions.

Cone distribution functions and quantiles for multivariate random variables

Set-valued quantiles for multivariate distributions with respect to a general convex cone are introduced which are based on a family of (univariate) distribution functions rather than on the joint distribution function. It is shown that these quantiles enjoy basically all the properties of univariate quantile functions. Relationships to families of univariate quantile functions and to depth functions are discussed. Finally, a corresponding Value at Risk for multivariate random variables as well as stochastic orders are introduced via the set-valued approach.

Total positivity in exponential families with application to binary variables

We study exponential families of distributions that are multivariate totally positive of order 2 (MTP2), show that these are convex exponential families, and derive conditions for existence of the MLE. Quadratic exponential familes of MTP2 distributions contain attractive Gaussian graphical models and ferromagnetic Ising models as special examples. We show that these are defined by intersecting the space of canonical parameters with a polyhedral cone whose faces correspond to conditional independence relations. Hence MTP2 serves as an implicit regularizer for quadratic exponential families and…