A Stein characterisation of the generalized hyperbolic distribution

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 tdistribution 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.

Products of normal, beta and gamma random variables: Stein operators and distributional theory

“…Very recently, Arras et al [1] have obtained a N -th order Stein operator for the distribution of a linear combination of N independent gammas random variables, although their Fourier approach is very different to ours. Also, in recent years, second order operators involving f , f ′ and f ′′ have appeared in the literature for the Laplace [34], variance-gamma distributions [10], [14], generalized hyperbolic distributions [15] and the PRR family of [32]. One of the main contributions of this paper is an extension of the product normal Stein operator (1.9) to mixed products of beta, gamma and normal random variables (see Propositions 2.3, 2.4 and 2.5).…”

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

“…If h is a bounded continuous function and p ≤ −1, then the function defined by Remark 3.1 This result was claimed by Gaunt (see [2]) with α = E(X) by applying Proposition 1 of [8]. The only slight mistake is that τ is not a polynomial function of degree one as in [8].…”

“…This enables us to apply Theorem 2.1 to retrieve the following Stein characterization of the GIG distribution given in [4] and [2]:…”

“…The aim of this paper is to solve the Stein equation and derive a bound of the solution for the Kummer distribution (which is new) and for the generalized inverse Gaussian distribution (which has been done in [2], but there was a little mistake in the bound of the solution).…”

“…Also a bound was obtained for the solution under the condition that the function τ be a decreasing linear function. But since this linearity condition does not hold in the GIG case, the bound given by [2] has to be slightly corrected. This is done in Theorem 3.1 after recalling the general framework of Schoutens [8] and adapting it to the cases where τ is decreasing but not necessarily linear.…”