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.

翻译：本文考虑将多元对称拉普拉斯分布推广至矩阵变元情形。对称拉普拉斯分布是正态分布的尺度混合。针对多元与矩阵变元对称拉普拉斯分布，本文提出了其参数的极大似然估计量，由于密度函数包含第三类修正贝塞尔函数，这些估计量无法显式求得。因此，采用EM算法求解极大似然估计量。矩阵变元对称拉普拉斯分布的参数及其极大似然估计量在正乘性常数意义下定义，其克罗内克积具有唯一性。文中给出了极大似然估计量存在的条件，并讨论了估计量的稳定性。通过模拟数据，分析了不同样本量下估计量克罗内克积的经验偏差与离散程度。