Stein's density approach and information inequalities

Abstract

We provide a new perspective on Stein's so-called density approach by introducing a new operator and characterizing class which are valid for a much wider family of probability distributions on the real line. We prove an elementary factorization property of this operator and propose a new Stein identity which we use to derive information inequalities in terms of what we call the "generalized Fisher information distance". We provide explicit bounds on the constants appearing in these inequalities for several important cases. We conclude with a comparison between our results and known results in the Gaussian case, hereby improving on several known inequalities from the literature.