Stein’s method for comparison of univariate distributions

Abstract

We propose a new general version of Stein's method for univariate distributions. In particular we propose a canonical definition of the Stein operator of a probability distribution which is based on a linear difference or differential-type operator. The resulting Stein identity highlights the unifying theme behind the literature on Stein's method (both for continuous and discrete distributions). Viewing the Stein operator as an operator acting on pairs of functions, we provide an extensive toolkit for distributional comparisons. Several abstract approximation theorems are provided. Our approach is illustrated for comparison of several pairs of distributions : normal vs normal, sums of independent Rademacher vs normal, normal vs Student, and maximum of random variables vs exponential, Fréchet and Gumbel.

First-order covariance inequalities via Stein’s method

“…The backbone of the present paper consists in the approach from [50,53,62]. Before introducing these results, we fix the notations.…”

“…Our first definitions come from [53]. We first define dom(p, ∆ ) as the collection of f : IR → IR such that f p ∈ dom(∆ ).…”

A Stein characterisation of the generalized hyperbolic distribution

“…Over the years, the method has been adapted to many other probability distributions, such as the Poisson [9], gamma [26,32], exponential [7,36] and Laplace distribution [14,39], and has been applied to a wide range of applications, including random matrix theory [33], random graph theory [3], urn models [13,28,38], goodness-of-fit statistics [27,26] and statistical physics [8,19]. For an overview of the current literature see [31]. In particular, the method has recently been extended to the variance-gamma distribution; see [21] and [20].…”

“…The fact that (1.5) is a second order differential equation in f , f ′ and f ′′ means that the standard generator [2,29] and density [45] methods (the scope of the density method has recently been extended by [31]) do not easily lend themselves to deriving this Stein equation. (Note that a direct application of the density method would lead to a first order Stein equation with complicated coefficients involving the modified Bessel function of the second kind, which would most likely be intractable in applications.)…”

“…To arrive at our characterisation of the GH distribution that leads to (1.5), we note a second order homogeneous differential equation that the density (1.4) satisfies and then exploit the duality between Stein equations and differential equations of densities to obtain our Stein equation. This is one the first explicit examples in the literature of this technique being used to derive a Stein equation, but see [31], Section 3.6 for a discussion and example of this approach.…”