Poisson approximation and the Chen-Stein method. With comments and a rejoinder by the authors

DOI10.1214/ss/1177012015zbMath0955.62542OpenAlexW1558717344WikidataQ56430004 ScholiaQ56430004MaRDI QIDQ1596002

Larry Goldstein, Richard Arratia, Louis Gordon

Publication date: 7 February 2001

Published in: Statistical Science (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1214/ss/1177012015

Mathematics Subject Classification ID

Asymptotic distribution theory in statistics (62E20)

Related Items (88)

Stein's method and birth-death processes ⋮ Compound Poisson approximation: A user's guide ⋮ A bound for the distribution of the sum of discrete associated or negatively associated random variables. ⋮ Compound Poisson approximation in total variation ⋮ Multivariate Poisson and Poisson process approximations with applications to Bernoulli sums and -statistics ⋮ Criteria for Poisson process convergence with applications to inhomogeneous Poisson-Voronoi tessellations ⋮ A Poisson approximation for the number of \(k\)-matches ⋮ Kendall's tau-type rank statistics in genome data. ⋮ Gamma-distribution order statistics, maximal multinomial frequency and randomization de\-signs ⋮ Stein's method for concentration inequalities ⋮ Individually rational pure strategies in large games ⋮ Extremal lifetimes of persistent cycles ⋮ Extremal Probability Bounds in Combinatorial Optimization ⋮ Matrix models, Toeplitz determinants and recurrence times for powers of random unitary matrices ⋮ Stein's method, Palm theory and Poisson process approximation. ⋮ Poisson perturbations ⋮ Strong approximations for epidemic models ⋮ Compound Poisson process approximation for locally dependent real-valued random variables via a new coupling inequality ⋮ Compound Poisson approximation for unbounded functions on a group, with application to large deviations ⋮ Two-dimensional discrete scan statistics ⋮ Binomial approximation for sum of indicators with dependent neighborhoods ⋮ On the normal approximation for the distribution of the number of simple or compound patterns in a random sequence of multi-state trials ⋮ Stein’s method and Plancherel measure of the symmetric group ⋮ Networks of polynomial pieces with application to the analysis of point clouds and images ⋮ Translated Poisson approximation for Markov chains ⋮ Two-stage false discovery rate in microarray studies ⋮ Approximating dependent rare events ⋮ The compound Poisson approximation for a portfolio of dependent risks ⋮ Some results associated with the longest run in a strongly ergodic Markov chain ⋮ Approximations for a three dimensional scan statistic ⋮ Approximations for success run probabilities in Bernoulli trials ⋮ On the characterization of Brownian bridge measure on the pinned path space over a compact Riemannian manifold ⋮ Compound Poisson approximations for sums of 1-dependent random variables. I ⋮ Random feedback shift registers and the limit distribution for largest cycle lengths ⋮ A probability approximation framework: Markov process approach ⋮ Spectral Correlation Hub Screening of Multivariate Time Series ⋮ Stein's method for nonlinear statistics: a brief survey and recent progress ⋮ A non-uniform birthday problem with applications to discrete logarithms ⋮ A Poisson Approximation for an Occupancy Problem with Collisions ⋮ On Negative Binomial Approximation to k-Runs ⋮ Approximation of projections of random vectors ⋮ Fundamentals of Stein's method ⋮ Improved lower bounds on the total variation distance for the Poisson approximation ⋮ Approximating the extreme right-hand tail probability for the distribution of the number of patterns in a sequence of multi-state trials ⋮ Note on nonuniform estimate for compound Poisson approximations to \(2\)-runs ⋮ On the length of the longest run in a multi-state Markov chain. ⋮ Stein’s method of normal approximation for dynamical systems ⋮ New approximations for the distribution of the \(r\)-scan statistic ⋮ Generalized Coupon Collection: The Superlinear Case ⋮ Berry-Esseen bounds of normal and nonnormal approximation for unbounded exchangeable pairs ⋮ Joint distributions of numbers of occurrences of a discrete pattern and weak convergence of an empirical process for the pattern ⋮ Extreme values for characteristic radii of a Poisson-Voronoi tessellation ⋮ Approximating the distributions of runs and patterns ⋮ Comparing bacterial genomes from linear orders of patterns ⋮ Asymptotic distribution of Bernoulli quadratic forms ⋮ Constrained versions of Sauer's Lemma ⋮ Poisson, compound Poisson and process approximations for testing statistical significance in sequence comparisons ⋮ An accurate approximation for the expected site frequency spectrum in a Galton-Watson process under an infinite sites mutation model ⋮ Inequalities for rare events in time-reversible Markov chains. II ⋮ The matching, birthday and the strong birthday problem: a contemporary review ⋮ A non-uniform bound on Poisson approximation in somatic cell hybrid model ⋮ Entropy based constrained inference for some HDLSS genomic models: UI tests in a Chen-Stein perspective ⋮ Asymptotics for local maximal stack scores with general loop penalty function ⋮ Stein's method, Jack measure, and the Metropolis algorithm ⋮ On bounds in Poisson approximation for integer-valued independent random variables ⋮ Spin glasses and Stein's method ⋮ Consecutive covering arrays and a new randomness test ⋮ New non-uniform bounds on Poisson approximation for dependent Bernoulli trials ⋮ Some results on approximation for distributions of random sums by Poisson distribution ⋮ Approximation of sojourn-times via maximal couplings: motif frequency distributions ⋮ On maxima of stationary fields ⋮ Detection of sparse positive dependence ⋮ Poisson statistics of eigenvalues in the hierarchical Dyson model ⋮ A general study of extremes of stationary tessellations with examples ⋮ Nonoverlapping Clusters: Approximate Distribution and Application to Molecular Biology ⋮ Approximate probabilities for runs and patterns in i.i.d. and Markov-dependent multistate trials ⋮ Asymptotic distribution for the birthday problem with multiple coincidences, via an embedding of the collision process ⋮ The Second-Moment Phenomenon for Monochromatic Subgraphs ⋮ On the distribution of pure strategy equilibria in finite games with vector payoffs ⋮ Asymptotic analysis for extreme eigenvalues of principal minors of random matrices ⋮ Monochromatic subgraphs in randomly colored graphons ⋮ Poisson approximation with applications to stochastic geometry ⋮ Compound Poisson approximation of word counts in DNA sequences ⋮ Rate of Convergence to the Poisson Law of the Numbers of Cycles in the Generalized Random Graphs ⋮ Improved compound Poisson approximation for the number of occurrences of any rare word family in a stationary markov chain ⋮ Limit laws for large th-nearest neighbor balls ⋮ A multi-functional analyzer uses parameter constraints to improve the efficiency of model-based gene-set analysis ⋮ Stein meets Malliavin in normal approximation

This page was built for publication: Poisson approximation and the Chen-Stein method. With comments and a rejoinder by the authors