Poisson Approximation and the Chen-Stein Method
TL;DR: The Chen-Stein method of Poisson approximation is a powerful tool for computing an error bound when approximating probabilities using the Poisson distribution as discussed by the authors, in many cases, this bound may be given in terms of first and second moments alone.
Abstract: The Chen-Stein method of Poisson approximation is a powerful tool for computing an error bound when approximating probabilities using the Poisson distribution. In many cases, this bound may be given in terms of first and second moments alone. We present a background of the method and state some fundamental Poisson approximation theorems. The body of this paper is an illustration, through varied examples, of the wide applicability and utility of the Chen-Stein method. These examples include birthday coincidences, head runs in coin tosses, random graphs, maxima of normal variates and random permutations and mappings. We conclude with an application to molecular biology. The variety of examples presented here does not exhaust the range of possible applications of the Chen-Stein method.
Citations
More filters
Book•
01 Jan 2007TL;DR: The Erdos-Renyi random graphs model, a version of the CHKNS model, helps clarify the role of randomness in the distribution of values in the discrete-time world.
Abstract: 1. Overview 2. Erdos-Renyi random graphs 3. Fixed degree distributions 4. Power laws 5. Small worlds 6. Random walks 7. CHKNS model.
1,010 citations
Book•
04 Nov 2005
TL;DR: In this article, the authors provide an essential guide to managing modern financial risk by combining coverage of stochastic order and risk measure theories with the basics of risk management, including dependence concepts and dependence orderings.
Abstract: The increasing complexity of insurance and reinsurance products has seen a growing interest amongst actuaries in the modelling of dependent risks. For efficient risk management, actuaries need to be able to answer fundamental questions such as: Is the correlation structure dangerous? And, if yes, to what extent? Therefore tools to quantify, compare, and model the strength of dependence between different risks are vital. Combining coverage of stochastic order and risk measure theories with the basics of risk management and stochastic dependence, this book provides an essential guide to managing modern financial risk. * Describes how to model risks in incomplete markets, emphasising insurance risks. * Explains how to measure and compare the danger of risks, model their interactions, and measure the strength of their association. * Examines the type of dependence induced by GLM-based credibility models, the bounds on functions of dependent risks, and probabilistic distances between actuarial models. * Detailed presentation of risk measures, stochastic orderings, copula models, dependence concepts and dependence orderings. * Includes numerous exercises allowing a cementing of the concepts by all levels of readers. * Solutions to tasks as well as further examples and exercises can be found on a supporting website.
590 citations
TL;DR: This map represents the two–thirds point toward the goal of developing a mouse genetic map containing 6,000 SSLPs, and there is a significant underrepresentation of markers on the X chromosome.
Abstract: We have constructed a genetic map of the mouse genome containing 4,006 simple sequence length polymorphims (SSLPs). The map provides an average spacing of 0.35 centiMorgans (cM) between markers, corresponding to about 750 kb. Approximately 90% of the genome lies within 1.1 cM of a marker and 99% lies within 2.2 cM. The markers have an average polymorphism rate of 50% in crosses between laboratory strains. The markers are distributed in a relatively uniform fashion across the genome, although some deviations from randomness can be detected. In particular, there is a significant underrepresentation of markers on the X chromosome. This map represents the two–thirds point toward our goal of developing a mouse genetic map containing 6,000 SSLPs.
558 citations
Book•
04 Nov 2010
TL;DR: In this paper, Stein's method is used for non-linear statistics and multivariate normal approximations for independent random variables with moderate deviations, and a non-normal approximation for nonlinear statistics.
Abstract: Preface.- 1.Introduction.- 2.Fundamentals of Stein's Method.- 3.Berry-Esseen Bounds for Independent Random Variables.- 4.L^1 Bounds.- 5.L^1 by Bounded Couplings.- 6 L^1: Applications.- 7.Non-uniform Bounds for Independent Random Variables.- 8.Uniform and Non-uniform Bounds under Local Dependence.- 9.Uniform and Non-Uniform Bounds for Non-linear Statistics.- 10.Moderate Deviations.- 11.Multivariate Normal Approximation.- 12.Discretized normal approximation.- 13.Non-normal Approximation.- 14.Extensions.- References.- Author Index .- Subject Index.- Notation.
468 citations
Book•
26 Jun 2017TL;DR: This authoritative text draws on theoretical advances of the past twenty years to synthesize all aspects of Bayesian nonparametrics, from prior construction to computation and large sample behavior of posteriors, making it valuable for both graduate students and researchers in statistics and machine learning.
Abstract: Explosive growth in computing power has made Bayesian methods for infinite-dimensional models - Bayesian nonparametrics - a nearly universal framework for inference, finding practical use in numerous subject areas. Written by leading researchers, this authoritative text draws on theoretical advances of the past twenty years to synthesize all aspects of Bayesian nonparametrics, from prior construction to computation and large sample behavior of posteriors. Because understanding the behavior of posteriors is critical to selecting priors that work, the large sample theory is developed systematically, illustrated by various examples of model and prior combinations. Precise sufficient conditions are given, with complete proofs, that ensure desirable posterior properties and behavior. Each chapter ends with historical notes and numerous exercises to deepen and consolidate the reader's understanding, making the book valuable for both graduate students and researchers in statistics and machine learning, as well as in application areas such as econometrics and biostatistics.
458 citations
References
More filters
46,339 citations
Book•
[...]
01 Sep 1985
7,736 citations
Additional excerpts
...For applications and examples, see Barbour, 1982; Bollobas, 1985; Holst, 1986; Janson, 1986; Stein, 1986; Barbour, Holst and Janson, 1988; Heckman, 1988; Barbour and Holst, 1989; and Holst and Janson, 1990.)...
[...]
Book•
06 Apr 2011
TL;DR: In this paper, Doubly Stochastic Matrices and Schur-Convex Functions are used to represent matrix functions in the context of matrix factorizations, compounds, direct products and M-matrices.
Abstract: Introduction.- Doubly Stochastic Matrices.- Schur-Convex Functions.- Equivalent Conditions for Majorization.- Preservation and Generation of Majorization.- Rearrangements and Majorization.- Combinatorial Analysis.- Geometric Inequalities.- Matrix Theory.- Numerical Analysis.- Stochastic Majorizations.- Probabilistic, Statistical, and Other Applications.- Additional Statistical Applications.- Orderings Extending Majorization.- Multivariate Majorization.- Convex Functions and Some Classical Inequalities.- Stochastic Ordering.- Total Positivity.- Matrix Factorizations, Compounds, Direct Products, and M-Matrices.- Extremal Representations of Matrix Functions.
6,641 citations
TL;DR: In this article, an unbiased estimate of risk is obtained for an arbitrary estimate, and certain special classes of estimates are then discussed, such as smoothing by using moving averages and trimmed analogs of the James-Stein estimate.
Abstract: Estimation of the means of independent normal random variables is considered, using sum of squared errors as loss. An unbiased estimate of risk is obtained for an arbitrary estimate, and certain special classes of estimates are then discussed. The results are applied to smoothing by use of moving averages and to trimmed analogs of the James-Stein estimate. A suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate.
2,866 citations
"Poisson Approximation and the Chen-..." refers background in this paper
...…use a multidimensional version of equation (1) to recover and generalize Stein's (1956) remarkable result on the inadmissability of the normal mean in three or more dimensions (Hudson, 1978), or to study other questions arising in the estimation of the mean of a multivariate normal (Stein, 1981)....
[...]
Book•
20 Nov 2011
2,686 citations