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

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.