About the Stein equation for the generalized inverse Gaussian and Kummer distributions

TL;DR

This paper develops a Stein characterization for the Kummer distribution, extending the framework used for the generalized inverse Gaussian distribution, and provides bounds for the solutions under certain parameter conditions.

Contribution

It introduces a novel Stein characterization for the Kummer distribution and corrects a previous error in the related GIG distribution framework.

Findings

Derived a Stein equation for the Kummer distribution

Established bounds for the Stein solution under parameter conditions

Revisited and corrected the GIG distribution framework

Abstract

We propose a Stein characterization of the Kummer distribution on (0, $\infty$). This result follows from our observation that the density of the Kummer distribution satisfies a certain differential equation, leading to a solution of the related Stein equation. A bound is derived for the solution, under a condition on the parameters. The derivation of this bound is carried out using the same framework as in Gaunt 2017 [A Stein characterisation of the generalized hyper-bolic distribution. ESAIM: Probability and Statistics, 21, 303--316] in the case of the generalized inverse Gaussian distribution, which we revisit by correcting a minor error in the latter paper.