An algebra of Stein operators

Abstract

We build upon recent advances on the distributional aspect of Stein's method to propose a novel and flexible technique for computing Stein operators for random variables that can be written as products of independent random variables. We show that our results are valid for a wide class of distributions including normal, beta, variance-gamma, generalized gamma and many more. Our operators are kth degree differential operators with polynomial coefficients; they are straightforward to obtain even when the target density bears no explicit handle. As an application, we derive a new formula for the density of the product of k independent symmetric variance-gamma distributed random variables. -Tilman, Allée de la découverte 12, B-8000 Liège, Belgium yswan@uliege.be. variable of interest W and taking the supremum over all h in some class of functions H leads to the estimatewhere the final supremum is taken over all f h that solve (1). The third and final step of the method involves developing appropriate strategies for bounding the expectation on the right hand side of (2). This is of interest because many important probability metrics (such as the Kolmogorov and Wasserstein metrics) are of the form d H (L(W ), L(X)). Moreover, in many settings bounding the expectation IE[Af h (W )] is relatively tractable, and as a result Stein's method has found application in disciplines as diverse as random graph theory [5], number theory [22], statistical mechanics [13] and quantum mechanics [29]. We refer to the survey paper [37] as well as to the monographs [30,9] for a deeper look into some of the fruits of Charles Stein's seminal insights, particularly in the case where the target is the normal distribution. The linchpin of the method is the operator A whose properties are crucial to the success of the whole enterprise. In the sequel, we concentrate exclusively on differential Stein operators (some operators in the literature are integral or even fractional, see e.g. [43,3]) and adopt the following lax definition:Definition 1.1. A linear differential operator A acting on a class F of functions is a Stein operator for X if (i) Af ∈ L 1 (X) and (ii) IE [Af (X)] = 0 for all f ∈ F.There are infinitely many Stein operators for any given target distribution. For instance, if the distribution is known (even if only up to a normalizing constant) then the "canonical" theory from [26] applies, leading to entire families of operators. This approach provides natural first order polynomial operators e.g. for target distributions which belong to the Pearson family [38] or which satisfy a diffusive assumption [11,25]. In some cases, one may rather apply a duality argument. For instance the p.d.f. γ(x) = (2π) −1/2 e −x 2 /2 of the standard normal distribution satisfies the first order ODE γ ′ (x) + xγ(x) = 0 leading, by integration by parts, to the already mentioned operator Af (x) = f ′ (x) − xf (x). This is particularly useful for densities defined implicitely via ODEs. Such are by no means the only methods for deriving differential Stein ope...

Some new Stein operators for product distributions

“…In this paper we pursue the work begun in Gaunt [9] and Gaunt [11] concerning the following question : "given two independent random variables X and Y with Stein operators A X and A Y , can one find a Stein operator for Z = XY ?" More specifically, the present paper is a complement (sequel) to our paper Gaunt, Mijoule and Swan [13] where we developed an algebraic technique for finding Stein operators for products of independent random variables with polynomial Stein operators satisfying a technical condition. Let M (f ) = (x → xf (x)), D(f ) = (x → f (x)) and I be the identity operator.…”

“…The highest value of j such that a ij = 0 is called the order of the operator. In Gaunt, Mijoule and Swan [13] we provided a method for deriving operators under the technical as-sumption that # {j − i | a ij = 0} ≤ 2 (see Assumption 3 and Lemma 2.6 of Gaunt, Mijoule and Swan [13] for more details on this condition). For such random variables, Proposition 2.12 of Gaunt, Mijoule and Swan [13] gives a polynomial Stein operator for the product XY .…”

“…In Gaunt, Mijoule and Swan [13] we provided a method for deriving operators under the technical as-sumption that # {j − i | a ij = 0} ≤ 2 (see Assumption 3 and Lemma 2.6 of Gaunt, Mijoule and Swan [13] for more details on this condition). For such random variables, Proposition 2.12 of Gaunt, Mijoule and Swan [13] gives a polynomial Stein operator for the product XY . A number of classical random variables have Stein operators which satisfy this assumption, such as the N (0, σ 2 ) distribution with Stein operator σ 2 D − M , with others including the gamma, beta, and even some more exotic distributions such as the zero-mean symmetric variance-gamma distribution and PRR distribution of Pëkoz, Röllin and Ross [21].…”

“…In fact, even the non-centered normal distribution does not satisfy this assumption, as its Stein operator σ 2 D + µI − M instead satisfies # {j − i | a ij = 0} = 3. In Proposition 2.1, we shall address the natural problem of extending the result of Gaunt, Mijoule and Swan [13] to treat the product of two independent random variables satisfying this new assumption. Here we have only added one level of complexity in the operator; nevertheless, as we will see later on, it is sufficient to include the classical cases of non-centered normal and non-centered gamma, and a more general sub-family of the variance-gamma distributions.…”

Stein’s method and the distribution of the product of zero mean correlated normal random variables

“…Stein characterisations of probability distributions are most commonly used as part of Stein's method to derive distributional approximations, with powerful applications in random graph and network theory (Franceschetti and Meester [8]), convergence rates in classical asymptotic results in statistics (Anastasiou and Reinert [1], Gaunt, Pickett and Reinert [14]), Bayesian statistics (Ley, Reinert and Swan [18]) and statistical learning and inference (Gorham et al [15]); see the survey Ross [24] for a list of further application areas. However, recently Gaunt [10] and Gaunt, Mijoule and Swan [12] have found a novel application for Stein characterisations, in which they are used to establish formulas for PDFs of distributions that are too difficult to obtain via other methods. The basic approach, which we shall employ in this note, is to obtain a Stein characterisation of Z and then apply integration by parts to the characterising equation to deduce an ordinary differential equation (ODE) that the PDF must satisfy, from which we easily obtain the formula (1.1) for the density.…”