Stein’s density approach for multivariate continuous distributions

We provide a general framework for Stein’s method for multivariate continuous distributions. The approach gives a collection of Stein operators, among which we highlight score-Stein operators and kernel-Stein operators. We propose several new and known Stein identities and characterizations, including multivariate Student distributions and Wishart distributions, and introduce many new operators which we call standardizations. We discuss Stein kernels, provide a connection with mass transport as well as explicit formulas for elliptical distributions. Applications include estimates for the Wasserstein distance between nested densities as well as Stein kernel discrepancies which we illustrate on several examples.