Stein's method for the matrix normal distribution

This work presents the first systematic development of Stein's method for matrix distributions. We establish the basic essential ingredients of Stein's method for matrix normal approximation: we derive a generator-based Stein identity from a matrix Ornstein--Uhlenbeck diffusion with two-sided scales, provide an explicit semigroup representation for the solution of the Stein equation, and obtain regularity estimates for the solution. The new methodology is illustrated with three statistical applications, these being smooth Wasserstein distance bounds to quantify the matrix central limit theorem, a Wasserstein distance bound for the matrix normal approximation of the centered matrix $T$ distribution, and the derivation of Stein's method-of-moments estimators for scale parameters of the matrix normal distribution.