Coordinate-wise transformation of probability distributions to achieve a Stein-type identity

Abstract It is shown that for any given multi-dimensional probability distribution with regularity conditions, there exists a unique coordinate-wise transformation such the transformed satisfies Stein-type identity. A sufficient condition existence referred to as copositivity of distributions. The proof based on an energy minimization problem over totally geodesic subset Wasserstein space. result considered alternative Sklar’s theorem regarding copulas, and also interpreted generalization diagonal scaling theorem. identity applied rating multivariate data. numerical procedure piece-wise uniform densities provided. Some open problems are discussed.