On the optimality of Stein factors

TL;DR

This paper investigates the necessity of logarithmic terms in Stein factors used for distributional approximation, providing probabilistic examples that justify their presence and offering a new perspective on Stein equation solutions.

Contribution

It constructs probabilistic examples demonstrating that logarithmic terms in Stein factors are essential, not just artifacts, and offers a novel interpretation of Stein equation solutions.

Findings

Constructed probabilistic examples confirming the necessity of logarithmic terms.

Showed that known bounds on Stein factors are sharp and unavoidable.

Provided a new interpretation of solutions to Stein equations.

Abstract

The application of Stein's method for distributional approximation often involves so called Stein factors (also called 'magic factors') in the bound of the solutions to Stein equations. However, in some cases these factors contain additional (undesirable) logarithmic terms. It has been shown for many Stein factors that the known bounds are sharp and thus that these additional logarithmic terms cannot be avoided in general. However, no probabilistic examples have appeared in the literature that would show that these terms in the Stein factors are not just unavoidable artefacts, but that they are there for a good reason. In this article we close this gap by constructing such examples. This also leads to a new interpretation of the solutions to Stein equations.