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In June 2010, a conference, Probability Approximations and Beyond, was held at the National University of Singapore (NUS), in honor of pioneering mathematician Louis Chen. Chen made the first of several seminal contributions to the theory and application of Stein’s method. One of his most important contributions has been to turn Stein’s concentration inequality idea into an effective tool for providing error bounds for the normal approximation in many settings, and in particular for sums of random variables exhibiting only local dependence. This conference attracted a large audience that came to pay homage to Chen and to hear presentations by colleagues who have worked with him in special ways over the past 40+ years.

The papers in this volume attest to how Louis Chen’s cutting-edge ideas influenced and continue to influence such areas as molecular biology and computer science. He has developed applications of his work on Poisson approximation to problems of signal detection in computational biology. The original papers contained in this book provide historical context for Chen’s work alongside commentary on some of his major contributions by noteworthy statisticians and mathematicians working today.



Stein’s Method


Chapter 1. Couplings for Irregular Combinatorial Assemblies

When approximating the joint distribution of the component counts of a decomposable combinatorial structure that is ‘almost’ in the logarithmic class, but nonetheless has irregular structure, it is useful to be able first to establish that the distribution of a certain sum of non-negative integer valued random variables is smooth. This distribution is not like the normal, and individual summands can contribute a non-trivial amount to the whole, so its smoothness is somewhat surprising. In this paper, we consider two coupling approaches to establishing the smoothness, and contrast the results that are obtained.
Andrew Barbour, Anna Pósfai

Chapter 2. Berry-Esseen Inequality for Unbounded Exchangeable Pairs

The Berry-Esseen inequality is well-established by the Stein method of exchangeable pair approach when the difference of the pair is bounded. In this paper we obtain a general result which can achieve the optimal bound under some moment assumptions. As an application, a Berry-Esseen bound of \(O(1/\sqrt{n})\) is derived for an independence test based on the sum of squared sample correlation coefficients.
Yanchu Chen, Qi-Man Shao

Chapter 3. Clubbed Binomial Approximation for the Lightbulb Process

In the so called lightbulb process, on days \(r=1,\ldots,n,\) out of n lightbulbs, all initially off, exactly r bulbs selected uniformly and independent of the past have their status changed from off to on, or vice versa. With \(W_n\) the number of bulbs on at the terminal time n and \(C_n\) a suitable clubbed binomial distribution,
$$ d_{{{\rm TV}}}(W_n,C_n) \leqslant 2.7314 \sqrt{n} e^{-(n+1)/3} \quad \hbox{for all}\,n \geqslant 1. $$
The result is shown using Stein’s method.
Larry Goldstein, Aihua Xia

Chapter 4. Coverage of Random Discs Driven by a Poisson Point Process

Motivated by the study of large-scale wireless sensor networks, in this paper we discuss the coverage problem that a pre-assigned region is completely covered by the random discs driven by a homogeneous Poisson point process. We first derive upper and lower bounds for the coverage probability. We then obtain necessary and sufficient conditions, in terms of the relation between the radius r of the discs and the intensity \(\lambda\) of the Poisson process, in order that the coverage probability converges to 1 or 0 when \(\lambda\) tends to infinity. A variation of Stein-Chen method for compound Poisson approximation is well used in the proof.
Guo-Lie Lan, Zhi-Ming Ma, Su-Yong Sun

Chapter 5. On the Optimality of Stein Factors

The application of Stein’s method for distributional approximation often involves so-called Stein factors (also called magic factors) in the bound of the solutions to Stein equations. However, in some cases these factors contain additional (undesirable) logarithmic terms. It has been shown for many Stein factors that the known bounds are sharp and thus that these additional logarithmic terms cannot be avoided in general. However, no probabilistic examples have appeared in the literature that would show that these terms in the Stein factors are not just unavoidable artefacts, but that they are there for a good reason. In this article we close this gap by constructing such examples. This also leads to a new interpretation of the solutions to Stein equations.
Adrian Röllin

Related Topics


Chapter 6. Basic Estimates of Stability Rate for One-Dimensional Diffusions

In the context of one-dimensional diffusions, we present basic estimates (having the same lower and upper bounds with a factor of 4 only) for four Poincaré-type (or Hardy-type) inequalities. The derivations of two estimates have been open problems for quite some time. The bounds provide exponentially ergodic or decay rates. We refine the bounds and illustrate them with typical examples.
Mu-Fa Chen

Chapter 7. Trend Analysis of Extreme Values

In Dierckx and Teugels (Environmetrics 2:1–26) we concentrated on testing whether an instantaneous change occurs in the value of the extreme value index. This short article illustrates with an explicit example that in some cases the extreme value index seems to change gradually rather than instantaneously. To this end a moving Hill estimator is introduced. Further a change point analysis and a trend analysis are performed. With this last analysis it is investigated whether a linear trend appears in the extreme value index.
Goedele Dierckx, Jef Teugels

Chapter 8. Renormalizations in White Noise Analysis

Renormalization has been applied in many places by using a method fitting for each situation. In this report, we are in a position where a white noise \(\{ \dot B(t), t \in R^1 \}\) is taken to be a variable system of random functions \(\varphi (\dot B).\) With this setting, renormalization plays the role that lets \(\varphi (\dot B)\) become a generalized white noise functional, the notion of which has been well established in white noise theory.
Takeyuki Hida

Chapter 9. M-Dependence Approximation for Dependent Random Variables

The purpose of this paper is to describe the m-dependence approximation and some recent results obtained by using the m-dependence approximation technique. In particular, we will focus on strong invariance principles of the partial sums and empirical processes, kernel density estimation, spectral density estimation and the theory on periodogram. This paper is an update of, and a supplement to the paper “m-Dependent Approximation” by the authors in The International Congress of Chinese Mathematicians (ICCM) 2007, Vol II, 720–734.
Zheng-Yan Lin, Weidong Liu

Chapter 10. Variable Selection for Classification and Regression in Large p, Small n Problems

Classification and regression problems in which the number of predictor variables is larger than the number of observations are increasingly common with rapid technological advances in data collection. Because some of these variables may have little or no influence on the response, methods that can identify the unimportant variables are needed. Two methods that have been proposed for this purpose are EARTH and Random forest (RF). This article presents an alternative method, derived from the GUIDE classification and regression tree algorithm, that employs recursive partitioning to determine the degree of importance of the variables. Simulation experiments show that the new method improves the prediction accuracy of several nonparametric regression models more than Random forest and EARTH. The results indicate that it is not essential to correctly identify all the important variables in every situation. Conditions for which this occurs are obtained for the linear model. The article concludes with an application of the new method to identify rare molecules in a large genomic data set.
Wei-Yin Loh
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