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 \emph{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.

This is a revised version of our paper "On a connection between Stein characterizations and Fisher information" (arXiv reference : arXiv:1111.2368). Essential changes have been made. Certain elements of the previous version remain relevant to the literature and have not been included in the present version, therefore we upload this as a new arXiv submission