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

Essomanda Konzou,Angelo Efoevi Koudou

doi:10.1051/ps/2020009

Abstract

We observe that the density of the Kummer distribution satisfies a certain differential equation, leading to a Stein characterization of this distribution and to a solution of the related Stein equation. A bound is derived for the solution and for its first and second derivatives. To provide a bound for the solution we partly use the same framework as in Gaunt 2017 [Stein, ESAIM: PS 21 (2017) 303–316] in the case of the generalized inverse Gaussian distribution, which we revisit by correcting a minor error. We also bound the first and second derivatives of the Stein equation in the latter case.

Highlights

For a > 0, b ∈ R, c > 0, the Kummer distribution with parameters a, b, c has density ka,b,c(x) =xa−1(1 + x)−a−be−cx, Γ(a)ψ(a, a − b + 1; c) (x > 0)where ψ is the confluent hypergeometric function of the second kind
For details on generalized inverse Gaussian distribution (GIG) and Kummer distributions see [6, 7, 10] and references therein, where one can see for instance that these distributions are involved in some characterization problems related to the so-called Matsumoto-Yor property
These two distributions are considered in the context of Stein’s method. This method introduced in [14] is a technique used to bound the error in the approximation of the distribution of a random variable of Keywords and phrases: Generalized inverse Gaussian distribution, Kummer distribution, Stein characterization

Summary

Introduction

Among many other results, [2] solved the GIG Stein equation and bounded the solution by using a general result obtained in [12] when the targeted distribution has a density g satisfying (s(x)g(x)) = τ (x)g(x). Schoutens [12] found a solution to the Stein equation (2.1) and established a bound for the solution, under the condition that the function τ be a decreasing linear function (which is the case for the so-called Pearson and Ord classes of distributions considered in [12]). Note that [13] obtained bounds for the solution of the Stein equation and its first derivative in another general context where τ is not decreasing, under other assumptions that are not needed in the GIG and Kummer cases, where we obtain more explicit bounds, as shown in the two sections

About the Stein equation of the generalized inverse Gaussian distribution

About the Stein equation related to the Kummer distribution

Bound of the derivative

The GIG distribution as the law of a continued fraction