About the Stein equation for the generalized inverse Gaussian and Kummer distributions
Résumé
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 [A Stein characterisation of the generalized hyperbolic 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. We also bound the first and second derivatives of the Stein equation in the latter case.
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