Maximum Likelihood Estimation of the Parameters of Matrix Variate Symmetric Laplace Distribution

TL;DR

This paper extends the symmetric Laplace distribution to matrix variate cases, proposing MLEs via EM algorithm, analyzing their properties, and validating with simulated data.

Contribution

It introduces maximum likelihood estimators for the matrix variate symmetric Laplace distribution using EM, addressing the challenge of intractable density functions.

Findings

MLEs are defined up to a positive constant with a unique Kronecker product.

Conditions for MLE existence are provided.

Empirical bias and dispersion are analyzed through simulations.

Abstract

This paper considers an extension of the multivariate symmetric Laplace distribution to matrix variate case. The symmetric Laplace distribution is a scale mixture of normal distribution. The maximum likelihood estimators (MLE) of the parameters of multivariate and matrix variate symmetric Laplace distribution are proposed, which are not explicitly obtainable, as the density function involves the modified Bessel function of the third kind. Thus, the EM algorithm is applied to find the maximum likelihood estimators. The parameters and their maximum likelihood estimators of matrix variate symmetric Laplace distribution are defined up to a positive multiplicative constant with their Kronecker product uniquely defined. The condition for the existence of the MLE is given, and the stability of the estimators is discussed. The empirical bias and the dispersion of the Kronecker product of the estimators for different sample sizes are discussed using simulated data.