Matrix-Variate Row Exponential Normal Distribution: Properties and Maximum Likelihood Estimation

Matrix-valued data may exhibit heterogeneous row-specific effects that are not easily captured by a common latent mixing mechanism. To accommodate this feature while preserving a tractable conditional Gaussian structure, we introduce the Matrix-Variate Row Exponential-Normal (MVREN) distribution, based on a hierarchical normal mean-mixture representation. The model combines a location matrix, row-specific mean shifts driven by independent exponential latent variables, and matrix-variate Gaussian noise. Its main probabilistic properties are derived, including the stochastic representation, density function, conditional distributions, moments, and covariance structure. The density has a closed-form expression involving the multivariate normal cumulative distribution function. Parameter estimation is carried out through an Expectation–Conditional Maximization algorithm, in which the latent exponential variables are treated as missing data and each conditional maximization step has a closed-form update. Simulation studies examine the accuracy and stability of the estimation procedure across different sample sizes and parameter settings, while the Beijing air-quality application illustrates its performance with real matrix-valued pollutant data. Together, these results suggest that the MVREN distribution provides a practical framework for modeling matrix observations with row-specific heterogeneity and asymmetric behavior.