Fundamentals of Stein’s Method

Abstract

Chapter 2 lays out the foundations of Stein’s method. First the Stein characterization for the normal is shown, and then bounds on the Stein equation, that will be required throughout the treatment, are derived. The multivariate Stein equation for the normal, and its solution by the generator method, is also presented.

The basic ‘leave one out’ coupling considered in Chap. 1 is but one variation on the many ways in which variables close to the one of interest can enter the Stein equation. Four additional, and somewhat overlapping, basic methods for handling variables in the Stein equation are introduced in this chapter: the K-function approach, the original exchangeable pair method of Stein, and the zero bias and size bias transformations. Illustrations of the use of these methods are provided, as well as a number of examples, some of which will continue as themes throughout the book. The independent case serves as one important testing ground throughout.