A Stein characterisation of the generalized hyperbolic distribution

School of Mathematics, The University of Manchester, Manchester M13 9PL, UK.
robert.gaunt@manchester.ac.uk

Received: 26 April 2016
Revised: 8 December 2016
Accepted: 15 March 2017

Abstract

The generalized hyperbolic (GH) distributions form a five parameter family of probability distributions that includes many standard distributions as special or limiting cases, such as the generalized inverse Gaussian distribution, Student’s t-distribution and the variance-gamma distribution, and thus the normal, gamma and Laplace distributions. In this paper, we consider the GH distribution in the context of Stein’s method. In particular, we obtain a Stein characterisation of the GH distribution that leads to a Stein equation for the GH distribution. This Stein equation reduces to the Stein equations from the current literature for the aforementioned distributions that arise as limiting cases of the GH superclass.