Stein's Method and the Laplace Distribution

By John Pike, Haining Ren. 2012

66 citations recorded by the source corpus.

Using Stein's method techniques, we develop a framework which allows one to bound the error terms arising from approximation by the Laplace distribution and apply it to the study of random sums of mean zero random variables. As a corollary, we deduce a Berry-Esseen type theorem for the convergence of certain geometric sums. Our results make use of a second order characterizing equation and a distributional transformation which is related to zero-biasing. 1. Background and Introduction Beginning with the publication of Charles Stein's seminal paper Stein (1972), probabilists and statisticians have developed a wide range of techniques based on characterizing equations for bounding the distance between the distribution of a random variable X and that of a random variable Z having some specied

Subfields: Artificial Intelligence, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d9f13458a7fbc33b10090c919aa43d556656898e

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Citations in the corpus

Stein's method for comparison of univariate distributions (2014, methodology): “Outside of the Gaussian and Poisson frameworks, for univariate distributions the method has now also been shown to be effective for : exponential approximation [17, 74], Gamma approximation [64, 76, 65], binomial approximation [29], Beta approximation [40, 26], the asymptotics of rank distributions ”
On Stein's method for products of normal random variables and zero bias couplings (2013, methodology): “Since then, Stein’s method has been extended to many other distributions, such as the Poisson [5], beta [8], [25], gamma [21], [29], [31], exponential [4], [34], Laplace [9], [36] and variance-gamma [13], [19]; for an overview see Reinert [37].”
Variance-Gamma approximation via Stein's method (2013, unknown): “Page 7/33 ejp.ejpecp.org Variance-Gamma approximation via Stein’s method A similar argument (with the difference being that here p(x) = κν,β(−x)νeβxKν(−x)) shows that I2 = { κν,β limx↑0 2 νΓ(ν + 1)f(x), ν ≥ 0, κν,β limx↑0 2 ν−1Γ(ν + 1)f(x), −1/2 < ν < 0.”
Chi-square approximation by Stein's method with application to Pearson's statistic (2015, methodology): “[29] Pike, J. and Ren, H. Stein’s method and the Laplace distribution.”