Stein factors for absolutely continuous distributions with interval support and applications.

Abstract

The purpose of this work is to provide Stein factors for absolutely continuous distributions with interval support. It is divided into two parts. The first part provide the general theory that will eventually yield Stein factors whereas the second part consist in applications.

In the theory part, we basically follow the Stein’s method, i.e. give a Stein characterisation, the Stein equation and the Stein solution to finally give Stein factors. We first follow this procedure for the special case of the normal distribution and then for the more general case of absolutely continuous distributions with interval support.

In the application part, we first apply the general theory in the normal, beta, gamma and Student cases. In order to apply the factors obtained, we provide some theories enabling to use them to bound the Wasserstein distance; hence we give a theory for the general case as well as zero-bias, K-function and exchangeable pairs methods for the more specific normal case. In particular and among others, we find an upper bound of the rate of convergence of the Student distribution towards the Gaussian and also find upper bounds on the rates of convergence of the Central Limit Theorem in the exponential and Bernoulli cases. For each of these applications, we find bounds with the same order as the Berry-Esseen theorem.