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

Gaunt, Robert E.

doi:10.1214/16-bjps349

Cited by 33 publications

(48 citation statements)

References 32 publications

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“…The product gamma Stein operator (20) is in exact agreement with the one obtained by [19]. However, the Stein operators (19) and (21) differ slightly from those of [16,19], because they act on different functions. 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).…”

Section: Mixed Products Of Centered Normal and Gamma Random Variablessupporting

confidence: 72%

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 Variablessupporting

confidence: 72%

An algebra of Stein operators

Gaunt¹,

Mijoule

Swan

2019

Journal of Mathematical Analysis and Applications

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“…As the Rayleigh distribution is a special case of the generalized gamma distribution, the following lemma follows as a special case of Proposition 2.3 of [16]. Beta(1, n−1).…”

Section: )mentioning

confidence: 97%

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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“…However, as Stein characterisations characterise probability distributions, we can use them to infer various distributional properties, such as moments of distributions (see [21]), moment generating and characteristic functions (see [24] and [21]), and formulas for probability density functions (see [25], in which a new formula was established for the density of the distribution of the mixed product of independent beta, gamma and centred normal random variables). We consider one such example here.…”

Section: Applications Of Proposition 31mentioning

confidence: 99%

“…Such Stein equations were uncommon in the literature, although recently [21,37,39] have obtained second order Stein equations. In fact, n-th order Stein equations have recently been obtained for the product of n independent beta, gamma and central normal random variables [24,25] and general linear combinations of gamma random variables [1].…”

Section: Introductionmentioning

confidence: 99%

A Stein characterisation of the generalized hyperbolic distribution

Gaunt

2017

ESAIM: PS

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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.

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

Cited by 33 publications

References 32 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

A Stein characterisation of the generalized hyperbolic distribution

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