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Three general approaches to Stein's method

2005; Linguagem: Inglês

10.1142/9789812567680_0004

1793-0758

Gesine Reinert,

Topological and Geometric Data Analysis

Lecture Notes Series, Institute for Mathematical Sciences, National University of SingaporeAn Introduction to Stein's Method, pp. 183-221 (2005) No AccessThree general approaches to Stein's methodGesine ReinertGesine ReinertDepartment of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UKhttps://doi.org/10.1142/9789812567680_0004Cited by:31 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: Stein's method is in no way restricted to the particular settings discussed in the previous chapters: normal, Poisson and compound Poisson approximation for random variables, and Poisson point process approximation. This chapter is intended to illustrate that the method can be very generally exploited, and that one can set up and apply Stein's method in a wide variety of contexts. Three main approaches for achieving this are described: the generator method, density equations and coupling equations. These are applied to chi-squared approximation, to weak laws of large numbers in general spaces, including spaces of measures, and to discrete Gibbs distributions. The S-I-R epidemic model, which is discussed in some detail, serves to illustrate the extent of the possibilities. FiguresReferencesRelatedDetailsCited By 31Stein's Method Meets Computational Statistics: A Review of Some Recent DevelopmentsAndreas Anastasiou, Alessandro Barp, François-Xavier Briol, Bruno Ebner and Robert E. Gaunt et al.1 Feb 2023 | Statistical Science, Vol. 38, No. 1Quantitative two-scale stabilization on the Poisson spaceRaphaël Lachièze-Rey, Giovanni Peccati and Xiaochuan Yang1 Aug 2022 | The Annals of Applied Probability, Vol. 32, No. 4Approximations related to the sums of m-dependent random variablesAmit N. Kumar, Neelesh S. Upadhye and P. Vellaisamy1 Jun 2022 | Brazilian Journal of Probability and Statistics, Vol. 36, No. 2On discrete Gibbs measure approximation to runsA. N. Kumar and N. S. Upadhye26 May 2020 | Communications in Statistics - Theory and Methods, Vol. 51, No. 5Stein's method for the Poisson–Dirichlet distribution and the Ewens sampling formula, with applications to Wright–Fisher modelsHan L. Gan and Nathan Ross1 Apr 2021 | The Annals of Applied Probability, Vol. 31, No. 2Stein's method for functions of multivariate normal random variablesRobert E. Gaunt1 May 2020 | Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Vol. 56, No. 2Approximating stationary distributions of fast mixing Glauber dynamics, with applications to exponential random graphsGesine Reinert and Nathan Ross1 Oct 2019 | The Annals of Applied Probability, Vol. 29, No. 5Hitting time and mixing time bounds of Stein's factorsMichael C.H. Choi1 Jan 2018 | Electronic Communications in Probability, Vol. 23, No. noneGeneralized gamma approximation with rates for urns, walks and treesErol A. Peköz, Adrian Röllin and Nathan Ross1 May 2016 | The Annals of Probability, Vol. 44, No. 3Functional Poisson approximation in Kantorovich–Rubinstein distance with applications to U-statistics and stochastic geometryLaurent Decreusefond, Matthias Schulte and Christoph Thäle1 May 2016 | The Annals of Probability, Vol. 44, No. 3Rates of Convergence in Normal Approximation Under Moment Conditions Via New Bounds on Solutions of the Stein EquationRobert E. Gaunt13 June 2014 | Journal of Theoretical Probability, Vol. 29, No. 1Quantitative stable limit theorems on the Wiener spaceIvan Nourdin, David Nualart and Giovanni Peccati1 Jan 2016 | The Annals of Probability, Vol. 44, No. 1Stein's method, logarithmic Sobolev and transport inequalitiesMichel Ledoux, Ivan Nourdin and Giovanni Peccati30 January 2015 | Geometric and Functional Analysis, Vol. 25, No. 1Stein's method of exchangeable pairs for the Beta distribution and generalizationsChristian Döbler1 Jan 2015 | Electronic Journal of Probability, Vol. 20, No. noneNonnormal Small Jump Approximation of Infinitely Divisible DistributionsZhiyi Chi22 February 2016 | Advances in Applied Probability, Vol. 46, No. 4Variance-Gamma approximation via Stein's methodRobert Gaunt1 Jan 2014 | Electronic Journal of Probability, Vol. 19, No. noneStein's Method for the Beta Distribution and the Pólya-Eggenberger UrnLarry Goldstein and Gesine Reinert30 January 2018 | Journal of Applied Probability, Vol. 50, No. 4Degree asymptotics with rates for preferential attachment random graphsErol A. Peköz, Adrian Röllin and Nathan Ross1 Jun 2013 | The Annals of Applied Probability, Vol. 23, No. 3Chaos of a Markov operator and the fourth moment conditionM. Ledoux1 Nov 2012 | The Annals of Probability, Vol. 40, No. 6Stein's Method and Stochastic OrderingsFraser Daly, Claude Lefèvre and Sergey Utev4 January 2016 | Advances in Applied Probability, Vol. 44, No. 2References7 February 2012Fundamentals of Stein's methodNathan Ross1 Jan 2011 | Probability Surveys, Vol. 8, No. noneInvariance principles for homogeneous sums: Universality of Gaussian Wiener chaosIvan Nourdin, Giovanni Peccati and Gesine Reinert1 Sep 2010 | The Annals of Probability, Vol. 38, No. 5Stein's method and Normal approximation of Poisson functionalsG. Peccati, J. L. Solé, M. S. Taqqu and F. Utzet1 Mar 2010 | The Annals of Probability, Vol. 38, No. 2Stein's Method and Stochastic Analysis of Rademacher FunctionalsGesine Reinert, Ivan Nourdin and Giovanni Peccati1 Jan 2010 | Electronic Journal of Probability, Vol. 15, No. noneA Three-Parameter Binomial ApproximationErol A. Peköz, Adrian Röllin, Vydas Čekanavičius and Michael Shwartz14 July 2016 | Journal of Applied Probability, Vol. 46, No. 4Stein's method and exact Berry–Esseen asymptotics for functionals of Gaussian fieldsIvan Nourdin and Giovanni Peccati1 Nov 2009 | The Annals of Probability, Vol. 37, No. 6Multivariate normal approximation with Stein's method of exchangeable pairs under a general linearity conditionGesine Reinert and Adrian Röllin1 Nov 2009 | The Annals of Probability, Vol. 37, No. 6Stein's method on Wiener chaosIvan Nourdin and Giovanni Peccati20 June 2008 | Probability Theory and Related Fields, Vol. 145, No. 1-2Stein's Method and Characters of Compact Lie GroupsJason Fulman5 December 2008 | Communications in Mathematical Physics, Vol. 288, No. 3Upper Bounds for Stein-Type OperatorsFraser Daly1 Jan 2008 | Electronic Journal of Probability, Vol. 13, No. none An Introduction to Stein's MethodMetrics History PDF download