Maximum Likelihood Estimation in the Multivariate and Matrix Variate Symmetric Laplace Distributions through Group Actions

TL;DR

This paper investigates maximum likelihood estimation for multivariate and matrix variate symmetric Laplace distributions using group actions, linking stability and likelihood properties, and addressing non-exponential family challenges.

Contribution

It introduces a novel approach connecting MLE problems to norm minimization over groups for these distributions, which are not in the exponential family.

Findings

MLE problems related to norm minimization over groups

Established connection between data stability and likelihood function properties

Provided theoretical insights into non-exponential family distributions

Abstract

In this paper, we study the maximum likelihood estimation of the parameters of the multivariate and matrix variate symmetric Laplace distributions through group actions. The multivariate and matrix variate symmetric Laplace distributions are not in the exponential family of distributions. We relate the maximum likelihood estimation problems of these distributions to norm minimization over a group and build a correspondence between stability of data with respect to the group action and the properties of the likelihood function.