Maximum Likelihood Estimation in the Multivariate and Matrix Variate Symmetric Laplace Distributions through Group Actions
Pooja Yadav, Tanuja Srivastava
TL;DR
This work addresses maximum likelihood estimation for multivariate and matrix variate symmetric Laplace distributions, which are not in the exponential family, by leveraging group actions and invariance. It reformulates ML as a norm-minimization over a group via a complete-data representation with Y = √W Z and Ψ = A^T A, linking estimation to optimizing over orbits and their stability. The key contributions are (i) casting the Laplace models as group models, (ii) proving the ML problem is equivalent to norm minimization on the group, and (iii) establishing a rigorous stability-based criterion (unstable/semistable/polystable/stable) that governs the existence and uniqueness of MLEs through Kempf–Ness theory. These insights extend to the matrix variate case and offer a principled, invariant-theoretic framework for ML in non-exponential family settings with practical implications for computation via two-step optimization over the group. Overall, the paper connects geometric invariant theory with statistical estimation, providing conditions for MLE existence/uniqueness and guiding efficient algorithmic approaches.
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.