STEIN’S METHOD AND APPLICATIONS:
A PROGRAM IN HONOR OF CHARLES STEIN
(28 July - 31 August 2003)
~ Abstracts ~
Normal approximation for sums of
i.i.d. random variables
Charles Stein, Stanford University
Using an identity of Bolthausen [1984] based on an exchangeable pair, I shall develop some identities related to the Berry-Esseen theorem for sums of independent, identically distributed random variables.
[1984] An estimate of the remainder in a combinatorial central limit theorem. Zeitschrift f"ur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 66, 379-386.
Parameter estimation for highly
structured systems
Adrian Baddeley, University of Western Australia
A highly structured stochastic system, such as a Markov random field or conditional independence model, can be represented as the stationary distribution of a Markov chain. Typically this representation is used as the basis of Markov Chain Monte Carlo methods for the system. However, it can also be used for Stein's method. Furthermore, the generator of this chain is a ready-made source of unbiased estimating equations for the model parameters. Special cases include maximum likelihood, maximum pseudolikelihood, the Takacs-Fiksel method for spatial point processes, and the reduced sample estimator of lifetime distribution from right-censored data.
Convergence rate for CLT of maxima in
cubes
Zhidong Bai, National University of Singapore
Some applications of Stein's
method
A. D. Barbour, Universitaet Züerich
Stein's method has proved to be an extremely versatile tool for establishing distributional approximations. In this talk, we describe three settings of apparently quite different kinds, in which Stein's method has nonetheless turned out to play a significant part in the analysis: `small world' networks, (logarithmic) combinatorial structures and biological metapopulation models. In each case, Stein's method emerged by accident, rather than by design, illustrating its widespread usefulness.
Mathematical modelling of parasitic
diseases
A. D. Barbour, Universitaet Züerich
Mathematical arguments have been used to help in fighting disease ever since the work of Daniel Bernoulli in the early 1700’s. A common theme nowadays is to use mathematical models to predict the efficacy of competing public health measures, such as various immunization strategies, in relation to their probable cost. In this talk, the speaker illustrates that modelling can be effective even at a more basic level. A pair of simple differential equations was used to make a discovery about the natural history of a widespread parasitic infection.
Stein's method and Palm distributions
Tim Brown, Australian National University
The probabilistic approach to Stein's method for distributions on the non-negative integers (using reversible Markov birth-death processes) generalises to point processes. It extends to what are often called mixed sample processes, processes in which the number of points on the whole space has an arbitrary distribution . Conditional on this number the points are distributed in an independent and identical way. These processes include the Poisson, binomial and negative binomial on compact spaces. Such processes are simple examples of a general class called polynomial birth-death processes. The general class can be used to refine the Poisson approximation to a binomial process using probability distributions in contrast to expansions using Stein's method which are typically not approximations by probability distributions. There are interesting connections for these processes with the point process theory of Palm distributions, all this brought into being from Stein's method.
Stein’s method, concentration
inequalities and normal approximation
Louis Chen, National University of Singapore
Concentration inequalities in normal approximation using Stein’s method correspond to the smoothing lemma in the Fourier analytic method. They provide bounds on the discrepancy between a distribution and its translations. The discrepancy could be Kolmogorov or total variation distance. We discuss the role such inequalities play in two kinds of normal approximation using Stein’s method. The first concerns uniform and non-uniform bounds on the difference between the distribution function of a sum of dependent random variables and the standard normal distribution function. The second, which uses coupling, is about normal approximation in some kind of total variation for sums of independent integer-valued random variables.
Applications of Stein’s method in
appearances of attributes
Ourania Chryssaphinou, University of Athens
Generalizing the concept of attribute as it has been analyzed by W. Feller we study the number of renewal appearances of general attributes. Using Stein’s method we obtain limit results of Poisson type under quite general conditions. We present certain applications concerned known models, their generalization and new ones.
Random variables defined on
random binary search trees
Luc Devroye, McGill University
Several random variables defined on random binary search trees have normal limit laws that can be obtained by Stein's method. We investigate in particular sums of functions of subtrees, and obtain universally applicable laws. Special cases include the number of leaves, the number of subtrees with a given pattern, the product of the sizes of all subtrees, and the sum of the heights of all subtrees.
Historical roots of Stein's
method
Persi Diaconis, Stanford University
Stein's Method developed over many years. I will examine forerunners such as Lindeberg's approach, Stein's early efforts using characterisitic functions for the combinatorial limit theorem and end with a review of original paper.
The Search for randomness
(Public lecture)
Persi Diaconis, Stanford University
The speaker will discuss some of our most primitive examples of random phenomena: tossing a coin, rolling dice and shuffling cards. While common practice can produce randomness, usually a close look shows that it just isn't so.
Stein's method for Gibbs
measures
Peter Eichelsbacher, Fakultaet fuer Mathematik, Ruhr-Universitaet
Bochum, Germany
Stein's method provides a way of finding approximations to a distribution of a random variable and gives at the same time estimates of the approximation error involved. We provide Stein's method for the class of Gibbs measures with a density eV where V is the energy function. Combining the probabilistic view of Stein's method introduced by Barbour and the observation from Preston that Gibbs states are invariant measures of certain time-reversible spatial birth death processes we introduce the Stein equation and give Stein bounds. Using size bias couplings, we treat an example of Gibbs convergence for strongly correlated random variables due to Chayes and Klein.
Stein's method, Poisson and
compound Poisson approximations (Tutorial lectures)
Torkel Erhardsson, Royal Institute of Technology, Stockholm
We give an overview of the use of Stein's method for Poisson approximation (lectures 1-2) and compound Poisson approximation (lectures 3-4). We shall focus on basic results: the Stein equations and properties of their solutions (in particular the so-called magic factors), and how these are used to construct explicit error bounds for Poisson or compound Poisson approximations of sums of nonnegative integer valued random variables (the local and coupling approaches, monotone couplings, etc.) We give proofs for the most important results, as well as examples of applications. We shall also discuss some generalizations of the basic results, including Poisson-Charlier expansions, signed compound Poisson measure approximation, and compound Poisson approximation as a by-product of Poisson process and compound Poisson process approximation.
Stein's method, Markov renewal
point processes and strong memoryless times
Torkel Erhardsson, Royal Institute of Technology, Stockholm
Let W be the number of points in (0,t] of a stationary finite-state Markov renewal point process with marks falling into a rare subset B of the state space S. We want to find a good approximating compound Poisson distribution for L(W) and an explicit bound for the approximation error. We present a partial solution to the problem, obtained by expressing W as an integral with respect to an embedded renewal-reward process, and using Stein's method, Palm theory for point processes, and couplings. For the error bound to be small, S should contain at least one frequently visited state. We show how to embed the Markov renewal point process into another such process which has a frequently visited state, using auxiliary random variables called strong memoryless times. As an illustrative example, we consider the number of points in (0,t] of a stationary renewal process.
Normal approximation for hierarchical
sequences
Larry Goldstein, University of Southern California
Asymptotic spectral distributions
via Stein equations
Friedrich Götze, Bielefeld University
We show the asymptotic Wigner distribution of eigenvalues of random matrices distributed according to a martingale type stochastic model for the coefficients, which includes the Wigner-Ensemble. We use a Stein equation with singularities at the boundary of the compact support to prove the result and investigage some applications. Further possible applications of Stein's method will be discussed.
Concise rates of convergence in the
Studentised central limit theorem
Peter Hall, Australian National University
The central limit theorem for the Studentised mean has been of interest since Gosset first considered randomly normalised statistics almost 100 years ago. Early work, for example based on Edgeworth expansions, was conducted under the assumption of sufficiently many finite moments. Much more recently it has been been shown that moment conditions do not directly affect the validity of the central limit theorem for Student's t statistic. Indeed, the Studentised mean has an asymptotic standard normal distribution if and only if the sampling distribution is in the domain of attraction of the normal law. Assuming only this minimal condition we discuss leading-term expansions of the distribution of the Studentised mean. Under additional conditions these expansions approximate those of Edgeworth type. They give concise rates of convergence in the central limit theorem.
Comparing stationary distributions
of Markov Chains through Stein's method
Susan Holmes, Stanford University
I will show how Stein's method can be used to compare distances between distributions. Starting with tunable birth and death chains, I will show how we can choose the exchangeable pair and then compare the Stein operators, thus providing useful bounds on distances between distributions for which we don't necessarily know the Stein equation to start with.
Characterization of Brownian motion
on manifolds – an infinite-dimensional extension of Stein’s
idea
Elton P. Hsu, Northwestern University
We explain how Stein’s characterization of the standard normal random variable is related to an integration by parts formula in the path space over a Riemannian manifold.
This relation motivates the following result: Brownian motion on a Riemannian manifold is uniquely characterized by the integration by parts property for the gradient operator.
The central limit theorem for the
independence number for minimal spanning trees on random
points in the unit square
Sungchul Lee, Yonsei University
Fixed-domain asymptotic for a
subclass of Matérn-type Gaussian random fields
Wei-Liem Loh, National University of Singapore
Michael Stein (1989) proposed the Matérn-type Gaussian random fields as a very flexible class of models for computer experiments. This talk considers a subclass of these models that are exactly once mean square differentiable. In particular, the likelihood function is determined in closed-form and under mild conditions, the sieve maximum likelihood estimators for the parameters of the covariance function are shown to be weakly consistent with respect to fixed domain asymptotics.
Random geometric graphs and
random directed graphs
Mathew Penrose, University of Durham
Random geometric graphs are a natural alternative to classical random graph models, in which n vertices are independently randomly scattered in d-dimensional space (typically, uniformly over the unit square with d=2), with an edge included between nearby points. We consider topics such as giant component, clique number, and number of appearences of a given graph as a subgraph. These graphs are the subject of a recent monograph by the speaker.
If time permits, we shall also discuss recent work with Andrew Wade on directed nearest-neighbour graphs. In both cases, Stein's method is a useful tool.
Stein's method for chi-square
distributions, law of large numbers, and discrete
distributions from a Gibbs view point (Tutorial lectures)
Gesine Reinert, University of Oxford
In these tutorial lectures it shall be illustrated how Stein's method can be applied for distributional approximations in more generality. The first lecture will treat chi-square approximations, giving an application of the so-called generator approach to Stein's method. Afterwards this approach will be applied to the law of large numbers, and to convergence of empirical measures. Both the generator approach and a coupling characterization are put into use in the next part, considering discrete distributions in general. Examples, including an epidemic model, will be presented in the last lecture.
Application of Stein's method to
word matches in DNA sequences
Gesine Reinert, University of Oxford
Word count statistics are essential in many procedures for assessing statistical significance in DNA sequence comparison. Statistical large- sample approximations often can only be used in connection with bounds on the quality of the approximation. This approach will be applied to the analysis of the so-called D2 test statistic, counting the number of k-word matches between two random sequences. This statistic is used in BLAST.
Stein's method and Edgeworth expansions for
sums of dependent variables
Vladimir Rotar, San Diego State
University and the Central Economics and Mathematics Institute
of Russian Academy of Sciences
Stein's method and normal
approximation (Tutorial lectures)
Qi-Man Shao, National University of Singapore and
University of Oregon
We shall give an overview of the use of Stein’s method for normal approximation. We start with basic results on the Stein equations and their solutions and then prove several classical limit theorems to illustrate the beauty of Stein’s method. The focus will be on the ideas behind different approaches such as the concentration inequality approach, induction approach and exchangeable pair approach. We shall also discuss the uniform and non-uniform Berry-Esseen bounds for independent random variables as well as for locally dependent variables.
The Berry-Esseen bound for the
Student t-statistic via Stein's method
Qi-Man Shao, National University of Singapore and
University of Oregon
We show how the Stein method can be used to prove the Berry-Esseen bound for the Student t-statistic and for studentized statistics in general. The approach is based on some randomized concentration inequalities.
Aspects of the probability theory
of the MST and TSP
J. Michael Steele, University of Pennsylvania
Operator inequalities and their
applications
Sergey Utev, University of Nottingham
- How to find a good approximation on the stop-loss premiums of the first and second orders in the individual risk model with independent claims occurrences?
- How to measure the impact of dependence between claims occurrences?
- How to derive extremal properties of Rademacher functions or find the best constants in the Rosenthal inequality?
The Stein--Chen method, stochastic orderings and operator inequalities are employed to answer these and similar questions.
Stein's method and Poisson process
approximation (Tutorial lectures)
Aihua Xia, National University of Singapore and University
of Melbourne
We start with an introduction lecture on Poisson processes on the real line and then Poisson point processes on a locally compact complete separable metric space. The focus will be on important properties of Poisson point processes and how to characterize a Poisson point process. The second part will review the basics of Markov immigration-death processes, followed by the definition of Markov immigration-death point processes. We then explain how a Markov immigration-death point process evolves, establish its generator and find its stationary distribution. The final part concentrates on how to use the knowledge of Markov immigration-death point processes to construct Stein's method for Poisson process approximation with various applications.
Stein's method for compound Poisson
approximation via immigration-death processes
Aihua Xia, National University of Singapore and University
of Melbourne
Central limit theorems in
geometric probability
Joseph E. Yukich, Lehigh University
We establish general central limit theorems for measures induced by binomial and Poisson point processes in d-dimensional space. The limiting Gaussian field has a covariance structure which depends on the density of the point process. The general results are applied to deduce weak convergence of measures induced by random graphs (minimal spanning tree, nearest neighbor, Voronoi, and sphere of influence graph), random sequential packing models (ballistic deposition and spatial birth growth models), the process of maximal points of a random sample, and as well as to measures induced by spacing statistics. The talk is based on joint work with Y. Baryshnikov and M. Penrose.
A central limit theorem and large
deviation principle for decomposable random variables
Martin Raič, University of Ljubljana Slovenia
We present a central limit theorem with non-uniform bounds for sums of dependent random variables. The decay is sufficiently fast to allow us to derive large deviation principles. Our result is based on the concept of decomposable random variables, introduced by Barbour, Karoński and Ruciński. This approach to dependence has turned out to be applicable in various contexts, such as local dependence, random graphs and random permutations.
Dependent superpositions of
point processes
Dominic Schuhmacher, University of Zürich
We compare the distribution of a (possibly dependent) superposition of point processes to a Poisson process distribution. Using the method, we obtain upper bounds on the d_2 distance (a Wasserstein distance) between these distributions.
