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

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.