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This paper concerns the development of Stein's method for chi-square approximation and its application to problems in statistics. New bounds for the derivatives of the solution of the gamma Stein equation are obtained. These bounds involve…

Probability · Mathematics 2017-05-30 Robert E. Gaunt , Alastair Pickett , Gesine Reinert

We present a straightforward formulation of Stein's method for the semicircular distribution, specifically designed for the analysis of non-commutative random variables. Our approach employs a non-commutative version of Stein's heuristic,…

Probability · Mathematics 2024-12-03 Mario Díaz , Arturo Jaramillo

We use the Stein-Chen method to prove new explicit inequalities for the total variation, Wasserstein and local distances between the distribution of a random diagonal sum of a Bernoulli matrix and a Poisson distribution. Approximation…

Probability · Mathematics 2024-09-04 Bero Roos

We obtain explicit error bounds for the $d$-dimensional normal approximation on hyperrectangles for a random vector that has a Stein kernel, or admits an exchangeable pair coupling, or is a non-linear statistic of independent random…

Probability · Mathematics 2020-09-08 Xiao Fang , Yuta Koike

We propose a new version of Stein's method of exchangeable pairs, which, given a suitable exchangeable pair $(W,W')$ of real-valued random variables, suggests the approximation of the law of $W$ by a suitable absolutely continuous…

Probability · Mathematics 2015-10-21 Christian Döbler

The Stein Variational Gradient Descent method is a variational inference method in statistics that has recently received a lot of attention. The method provides a deterministic approximation of the target distribution, by introducing a…

Analysis of PDEs · Mathematics 2024-12-16 José A. Carrillo , Jakub Skrzeczkowski , Jethro Warnett

In this article, we develop Stein characterization for two-sided tempered stable distribution. Stein characterizations for normal, gamma, Laplace, and variance-gamma distributions already known in the literature follow easily. One can also…

Probability · Mathematics 2022-01-06 Kalyan Barman , N. S. Upadhye

In this article, we derive Stein's method for approximating a spatial random graph by a generalised random geometric graph, which has vertices given by a finite Gibbs point process and edges based on a general connection function. Our main…

Probability · Mathematics 2024-11-06 Dominic Schuhmacher , Leoni Carla Wirth

This article studies a general divide-and-conquer algorithm for approximating continuous one-dimensional probability distributions with finite mean. The article presents a numerical study that compares pre-existing approximation schemes…

We use Stein's method to obtain bounds on the rate of convergence for a class of statistics in geometric probability obtained as a sum of contributions from Poisson points which are exponentially stabilizing, i.e. locally determined in a…

Probability · Mathematics 2007-05-23 Mathew D. Penrose , J. E. Yukich

New bounds on the total variation distance between the law of integer valued functionals of possibly non-symmetric and non-homogeneous infinite Rademacher sequences and the Poisson distribution are established. They are based on a…

Probability · Mathematics 2017-07-26 Kai Krokowski

Simple inequalities are established for some integrals involving the modified Bessel functions of the first and second kind. In most cases, we show that we obtain the best possible constant or that our bounds are tight in certain limits. We…

Classical Analysis and ODEs · Mathematics 2018-02-09 Robert E. Gaunt

We provide a new perspective on Stein's so-called density approach by introducing a new operator and characterizing class which are valid for a much wider family of probability distributions on the real line. We prove an elementary…

Probability · Mathematics 2013-04-05 Christophe Ley , Yvik Swan

In a recent paper, Gaunt 2020 extended Stein's method to limit distributions that can be represented as a function $g:\mathbb{R}^d\rightarrow\mathbb{R}$ of a centered multivariate normal random vector $\Sigma^{1/2}\mathbf{Z}$ with…

Probability · Mathematics 2022-09-21 Robert E. Gaunt , Heather Sutcliffe

Motivated by a theorem of Barbour, we revisit some of the classical limit theorems in probability from the viewpoint of the Stein method. We setup the framework to bound Wasserstein distances between some distributions on infinite…

Probability · Mathematics 2018-07-30 Laure Coutin , Laurent Decreusefond

In this paper, a new method based on probability generating functions is used to obtain multiple Stein operators for various random variables closely related to Poisson, binomial and negative binomial distributions. Also, Stein operators…

Probability · Mathematics 2016-05-10 N. S. Upadhye , V. Cekanavicius , P. Vellaisamy

We obtain rates of convergence in limit theorems of partial sums $S_n$ for certain sequences of dependent, identically distributed random variables, which arise naturally in statistical mechanics, in particular, in the context of the…

Probability · Mathematics 2009-08-14 Peter Eichelsbacher , Matthias Löwe

In this paper we establish a multivariate exchangeable pairs approach within the framework of Stein's method to assess distributional distances to potentially singular multivariate normal distributions. By extending the statistics into a…

Probability · Mathematics 2010-04-06 Gesine Reinert , Adrian Röllin

We use a new method via $p$-Wasserstein bounds to prove Cram\'er-type moderate deviations in (multivariate) normal approximations. In the classical setting that $W$ is a standardized sum of $n$ independent and identically distributed…

Probability · Mathematics 2022-05-27 Xiao Fang , Yuta Koike

Variational Inference approximates an unnormalized distribution via the minimization of Kullback-Leibler (KL) divergence. Although this divergence is efficient for computation and has been widely used in applications, it suffers from some…

Machine Learning · Statistics 2022-07-28 Mingxuan Yi , Song Liu