Stein's Method for the Beta Distribution and the Pólya-Eggenberger Urn

Goldstein, Larry; Reinert, Gesine

doi:10.1017/s0021900200013875

Cited by 57 publications

(123 citation statements)

References 33 publications

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“…Indeed, the product normal Stein operator given in [16] isÃ X 1 ···Xn = σ 2 1 · · · σ 2 n DT n 0 − M , but multiplying through on the right by M yields (19). The same is true of the mixed product operator (21), which is equivalent to the mixed normal-gamma Stein operator of [19] multiplied on the right by M . We refer to Appendix A where this idea is expounded.…”

Section: Mixed Products Of Centered Normal and Gamma Random Variablesmentioning

confidence: 99%

“…We give a list of Stein operators for several classical probability distributions, in terms of the above operators. References for these Stein operators are as follows: normal [41], gamma [10,27], beta [11,21,38], Student's t [38], inverse-gamma [23], F -distribution (new to this paper), PRR [34], variance-gamma [15], and generalized gamma [19].…”

Section: A List Of Stein Operators For Continuous Distributionsmentioning

confidence: 99%

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An algebra of Stein operators

Gaunt¹,

Mijoule

Swan

2019

Journal of Mathematical Analysis and Applications

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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...

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Section: Mixed Products Of Centered Normal and Gamma Random Variablesmentioning

confidence: 99%

Section: A List Of Stein Operators For Continuous Distributionsmentioning

confidence: 99%

An algebra of Stein operators

Gaunt¹,

Mijoule

Swan

2019

Journal of Mathematical Analysis and Applications

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“…Proof. Define the operator T r by T r y(x) = xy ′ (x) + ry(x), r ∈ R. In this notation, the classical Stein operator for the Beta(1, n−1) distribution is given by A B n−1 y(x) = T 1 y(x)− xT n y(x) [5,22]. Let C n = B 1/2 n−1 and let g : (0, 1) → R by such that E|C n g ′ (C n )| < ∞,…”

Section: )mentioning

confidence: 99%

“…where here and throughout the paper g := g ∞ = sup x∈R |g(x)|, gives the Kolmogorov, Wasserstein and bounded Wasserstein distances, which we denote by d K , d W and d BW , respectively, as well as two smooth test function metrics, which we denote by d 2 and d 1,2 , respectively. The d 2 and d 1,2 and similar smooth test function metrics are often found in applications of Stein's method in which 'fast' convergence rates are sought, see, for example, [2,12,20,22]. Stein's method was adapted to the Laplace distribution by [37] (a number of their contributions are outlined in Section 2), and as an application they derived an explicit bound on the bounded Wasserstein distance between the distribution of S n and its limiting Laplace distribution.…”

Section: Introductionmentioning

confidence: 99%

Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein’s Method I

Gaunt

2018

J Theor Probab

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We use Stein's method to obtain explicit bounds on the rate of convergence for the Laplace approximation of two different sums of independent random variables; one being a random sum of mean zero random variables and the other being a deterministic sum of mean zero random variables in which the normalisation sequence is random. We make technical advances to the framework of Pike and Ren [37] for Stein's method for Laplace approximation, which allows us to give bounds in the Kolmogorov and Wasserstein metrics. Under the additional assumption of vanishing third moments, we obtain faster convergence rates in smooth test function metrics. As part of the derivation of our bounds for the Laplace approximation for the deterministic sum, we obtain new bounds for the solution, and it first two derivatives, of the Rayleigh Stein equation.where the test function h is real-valued. The second step is to solve (1.2) for f h and obtain suitable bounds for the solution. Finally, to approximate the distribution of a random variable of interest W by the target distribution q, one may evaluate both sides of (1.2) at W , take expectations and finally take the supremum of both sides over a class of functions H to obtain d H (W, Y ) := sup h∈H |E[h(W )] − E[h(Y )]| = sup h∈H E[Af h (W )].

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“…Thus, in order to bound |E(h(U)) − E(h(Z))| given h, its enough to find a solution f h of the Stein equation and to bound the left-hand side of the previous equation. The problem of solving the Stein equation for other distributions than the standard normal distribution and bounding the solution and its derivatives has been widely studied in the literature (see [3] among many others).…”

Section: Introductionmentioning

confidence: 99%

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

Konzou

Koudou

2020

ESAIM: PS

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We propose a Stein characterization of the Kummer distribution on (0, ∞). This result follows from our observation that the density of the Kummer distribution satisfies a certain differential equation, leading to a solution of the related Stein equation. A bound is derived for the solution, under a condition on the parameters. The derivation of this bound is carried out using 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 in the latter paper.

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Stein's Method for the Beta Distribution and the Pólya-Eggenberger Urn

Cited by 57 publications

References 33 publications

An algebra of Stein operators

An algebra of Stein operators

Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein’s Method I

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

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