New rates for exponential approximation and the theorems of Rényi and Yaglom

Peköz, Erol A.; Röllin, Adrian

doi:10.1214/10-aop559

Cited by 88 publications

(140 citation statements)

References 26 publications

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“…(2.16) Theorem 2.5 below gives general bounds for Laplace approximation involving the centered equilibrium transformation. Bounds (2.21) -(2.25) of the theorem are the Laplace analogues of the bounds of Theorem 2.1 of [34], which give Kolmogorov and Wasserstein distance bounds in terms the absolute difference between a random variable W and its W -equilibrium transformation. We additionally provide a bound in the weaker d 2 metric, which is used to obtain the O(p −1 ) bound (1.6) of Theorem 1.1.…”

Section: Stein's Methods For the Laplace Distributionmentioning

confidence: 97%

“…Stronger conditions were imposed on f by [37], but on examining the proof of their Theorem 3.2 it can be seen that the weaker conditions presented here are sufficient to ensure W L exists and is unique. We also refer the reader to [6] for a generalisation of (2.15) to all random variables W with finite second moment, and we note that the centered equilibrium distribution is itself the Laplace analogue of the equilibrium distribution that is used in Stein's method for exponential approximation by [34]. Some useful properties of the centered equilibrium transformation are collected in Section 3 of [37] and Proposition 4.6 of [17].…”

Section: Stein's Methods For the Laplace Distributionmentioning

confidence: 99%

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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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Section: Stein's Methods For the Laplace Distributionmentioning

confidence: 97%

Section: Stein's Methods For the Laplace Distributionmentioning

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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“…In a different form, the characterization given in Lemma 3.1 has successfully been applied for distributional approximations in the Curie-Weiss model [see Chatterjee and Shao (2011)] or the hitting times of Markov chains [see Peköz and Röllin (2011)]. For an overview, we refer to Section 13 in Chen et al (2011).…”

Section: The Density Approach Identitymentioning

confidence: 99%

Fixed point characterizations of continuous univariate probability distributions and their applications

Betsch

Ebner

2019

Ann Inst Stat Math

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By extrapolating the explicit formula of the zero-bias distribution occurring in the context of Stein's method, we construct characterization identities for a large class of absolutely continuous univariate distributions. Instead of trying to derive characterizing distributional transformations that inherit certain structures for the use in further theoretic endeavours, we focus on explicit representations given through a formula for the density-or distribution function. The results we establish with this ambition feature immediate applications in the area of goodness-of-fit testing. We draw up a blueprint for the construction of tests of fit that include procedures for many distributions for which little (if any) practicable tests are known.To illustrate this last point, we construct a test for the Burr Type XII distribution for which, to our knowledge, not a single test is known aside from the classical universal procedures.MSC 2010 subject classifications. Primary 62E10 Secondary 60E10, 62G10Over the last decades, Stein's method for distributional approximation has become a viable tool for proving limit theorems and establishing convergence rates. At it's heart lies the well-known Stein characterization which states that a real-valued random variable Z has a standard normal distribution if, and only if,holds for all functions f of a sufficiently large class of test functions. To exploit this characterization for testing the hypothesisof normality, where P X is the distribution of a real-valued random variable X, against general alternatives, Betsch and Ebner (2019b) used that (1.1) can be untied from the class of test functions with the help of the so-called zero-bias transformation introduced by Goldstein and Reinert (1997). To be specific, a real-valued random variable X * is said to have the X-zero-bias distri-bution ifholds for any of the respective test functions f . If EX = 0 and Var(X) = 1, the X-zero-bias distribution exists and is unique, and it has distribution function(1.3)By (1.1), the standard Gaussian distribution is the unique fixed point of the transformation P X → P X * . Thus, the distribution of X is standard normal if, and only if,where F X denotes the distribution function of X. In the spirit of characterization-based goodnessof-fit tests, an idea introduced by Linnik (1962), this fixed point property directly admits a new class of testing procedures as follows. Letting T X n be an empirical version of T X and F n the empirical distribution function, both based on the standardized sample, Betsch and Ebner (2019b) proposed a test for (1.2) based on the statisticwhere w is an appropriate weight function, which, in view of (1.4), rejects the normality hypothesis for large values of G n . As these tests have several desirable properties such as consistency against general alternatives, and since they show a very promising performance in simulations, we devote this work to the question to what extent the fixed point property and the class of goodness-of-fit procedures may be generalized to othe...

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“…For a detailed account of the method see the monograph [10] or the review article [40]. 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].…”

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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New rates for exponential approximation and the theorems of Rényi and Yaglom

Cited by 88 publications

References 26 publications

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

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

Fixed point characterizations of continuous univariate probability distributions and their applications

A Stein characterisation of the generalized hyperbolic distribution

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