Fundamentals of Stein's method

@article{Ross2011FundamentalsOS,
  title={Fundamentals of Stein's method},
  author={Nathan Ross},
  journal={Probability Surveys},
  year={2011},
  volume={8},
  pages={210-293}
}
  • Nathan Ross
  • Published 9 September 2011
  • Mathematics
  • Probability Surveys
This survey article discusses the main concepts and techniques of Stein's method for distributional approximation by the normal, Poisson, exponential, and geometric distributions, and also its relation to concentration of measure inequalities. The material is presented at a level accessible to beginning raduate students studying probability with the main emphasis on the themes that are common to these topics and also to much of the Stein's method literature. 

Stein's method for Borel approximation

We develop the tools necessary to use Stein's method for approximation by a Borel distribution. Two applications of these results are discussed. One is an explicit error bound in the approximation of

The Stein Method

Methodological aspects of the Stein method are exceptionally well discussed in the literature, and for more advanced applications the reader is advised to consult the books and papers referenced at

Archimedes, Gauss, and Stein

We discuss a characterization of the centered Gaussian distribution which can be read from results of Archimedes and Maxwell, and relate it to Charles Stein's well-known characterization of the same

Stein’s method and the distribution of the product of zero mean correlated normal random variables

  • Robert E. Gaunt
  • Mathematics
    Communications in Statistics - Theory and Methods
  • 2019
Abstract Over the last 80 years there has been much interest in the problem of finding an explicit formula for the probability density function of two zero mean correlated normal random variables.

Stein’s method via induction

Applying an inductive technique for Stein and zero bias couplings yields Berry-Esseen theorems for normal approximation with optimal rates in the Kolmogorov metric for two new examples. The

Approximating by convolution of the normal and compound Poisson laws via Stein’s method

We apply Stein’s method to estimate the closeness of the mixtures of distributions to the convolution of the normal and Poisson laws. To derive the Stein operator, we use a moment generating

Stein's Method Meets Computational Statistics: A Review of Some Recent Developments

Stein’s method compares probability distributions through the study of a class of linear operators called Stein operators. While mainly studied in probability and used to underpin theoretical

Stein's method for comparison of univariate distributions

We propose a new general version of Stein's method for univariate distributions. In particular we propose a canonical definition of the Stein operator of a probability distribution {which is based on

On Stein's method and mod-* convergence

Stein's method allows to prove distributional convergence of a sequence of random variables and to quantify it with respect to a given metric such as Kolmogorov's (a Berry-Ess\'een type theorem).

Normal approximation for associated point processes via Stein's method with applications to determinantal point processes

...

References

SHOWING 1-10 OF 65 REFERENCES

Stein's Method: Expository Lectures and Applications

TLDR
A review of Stein’s method applied to the case of discrete random variables and attempt to complete one of Stein's open problems, that of providing a discrete version for chapter 6 of his book.

Normal Approximation by Stein ’ s Method

The aim of this paper is to give an overview of Stein’s method, which has turned out to be a powerful tool for estimating the error in normal, Poisson and other approximations, especially for sums of

Stein's method for geometric approximation

  • E. Peköz
  • Mathematics
    Journal of Applied Probability
  • 1996
The Stein–Chen method for Poisson approximation is adapted to the setting of the geometric distribution. This yields a convenient method for assessing the accuracy of the geometric approximation to

Exponential Approximation by Stein's Method and Spectral Graph Theory

General Berry-Esseen bounds are developed for the exponential distri- bution using Stein's method and a new concentration inequality approach. As an application, a sharp error term is obtained for

An Introduction to Stein's Method

A common theme in probability theory is the approximation of complicated probability distributions by simpler ones, the central limit theorem being a classical example. Stein's method is a tool which

Stein ’ s method for concentration inequalities

We introduce a version of Stein’s method for proving concentration and moment inequalities in problems with dependence. Simple illustrative examples from combinatorics, physics, and mathematical

Poisson Approximation and the Chen-Stein Method

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

Applications of Stein's method for concentration inequalities

Stein’s method for concentration inequalities was introduced to prove concentration of measure in problems involving complex dependencies such as random permutations and Gibbs measures. In this

Stein couplings for normal approximation

In this article we propose a general framework for normal approximation using Stein's method. We introduce the new concept of Stein couplings and we show that it lies at the heart of popular

Normal Approximation by Stein's Method

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
...