Normal Approximation

-Fourier analysis, --

Since its introduction in 1972, Stein’s method has offered a completely novel way of evaluating the quality of normal approximations. Through its characterizing equation approach, it is able to provide approximation error bounds in a wide variety of situations, even in the presence of complicated dependence. Use of the method thus opens the door to the analysis of random phenomena arising in areas including statistics, physics, and molecular biology. Though Stein's method for normal approximation is now mature, the literature has so far lacked a complete self contained treatment. This volume contains thorough coverage of the method’s fundamentals, includes a large number of recent developments in both theory and applications, and will help accelerate the appreciation, understanding, and use of Stein's method by providing the reader with the tools needed to apply it in new situations. It addresses researchers as well as graduate students in Probability, Statistics and Combinatorics.

This book shows how quantitative central limit theorems can be deduced by combining two powerful probabilistic techniques: Stein's method and Malliavin calculus.

The series is devoted to the publication of high-level monographs and surveys which cover the whole spectrum of probability and statistics. The books of the series are addressed to both experts and advanced students.

Probability theory is one branch of mathematics that is simultaneously deep and immediately applicable in diverse areas of human endeavor. It is as fundamental as calculus. Calculus explains the external world, and probability theory helps predict a lot of it. In addition, problems in probability theory have an innate appeal, and the answers are often structured and strikingly beautiful. A solid background in probability theory and probability models will become increasingly more useful in the twenty-?rst century, as dif?cult new problems emerge, that will require more sophisticated models and analysis. Thisisa text onthe fundamentalsof thetheoryofprobabilityat anundergraduate or ?rst-year graduate level for students in science, engineering,and economics. The only mathematical background required is knowledge of univariate and multiva- ate calculus and basic linear algebra. The book covers all of the standard topics in basic probability, such as combinatorial probability, discrete and continuous distributions, moment generating functions, fundamental probability inequalities, the central limit theorem, and joint and conditional distributions of discrete and continuous random variables. But it also has some unique features and a forwa- looking feel.

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 makes this possible in a wide variety of situations. Traditional approaches, for example using Fourier analysis, become awkward to carry through in situations in which dependence plays an important part, whereas Stein's method can often still be applied to great effect. In addition, the method delivers estimates for the error in the approximation, and not just a proof of convergence. Nor is there in principle any restriction on the distribution to be approximated; it can equally well be normal, or Poisson, or that of the whole path of a random process, though the techniques have so far been worked out in much more detail for the classical approximation theorems.This volume of lecture notes provides a detailed introduction to the theory and application of Stein's method, in a form suitable for graduate students who want to acquaint themselves with the method. It includes chapters treating normal, Poisson and compound Poisson approximation, approximation by Poisson processes, and approximation by an arbitrary distribution, written by experts in the different fields. The lectures take the reader from the very basics of Stein's method to the limits of current knowledge.

This book was originally compiled for a course I taught at the University of Rochester in the fall of 1991, and is intended to give advanced graduate students in statistics an introduction to Edgeworth and saddlepoint approximations, and related techniques. Many other authors have also written monographs on this subject, and so this work is narrowly focused on two areas not recently discussed in theoretical text books. These areas are, first, a rigorous consideration of Edgeworth and saddlepoint expansion limit theorems, and second, a survey of the more recent developments in the field. In presenting expansion limit theorems I have drawn heavily 011 notation of McCullagh (1987) and on the theorems presented by Feller (1971) on Edgeworth expansions. For saddlepoint notation and results I relied most heavily on the many papers of Daniels, and a review paper by Reid (1988). Throughout this book I have tried to maintain consistent notation and to present theorems in such a way as to make a few theoretical results useful in as many contexts as possible. This was not only in order to present as many results with as few proofs as possible, but more importantly to show the interconnections between the various facets of asymptotic theory. Special attention is paid to regularity conditions. The reasons they are needed and the parts they play in the proofs are both highlighted.

An Introduction to Probability and Statistics An Introduction to Probability and Statistics, First Edition, guides the readers through basic probability and statistical methods along with graphs and tables and helps to analyse critically about various basic concepts. Written by two friends i.e. Dr. Arun Kaushik and Dr. Rajwant K. Singh, this book introduces readers with no or very little prior knowledge in probability or statistics to a thinking process to help them obtain the best solution to a posed situation. It provides lots of examples for each topic discussed, and examples are covered from the medical field giving the reader more exposure in applying statistical methods to different situations. This text contains an enhanced number of exercises and graphical illustrations to motivate the readers and demonstrate the applicability of probability and statistical inference in a vast variety of human activities. Each section includes relevant proofs where ever need arises, followed by exercises with some useful clues to their solutions. Furthermore, if the need arises then the detailed solutions to all exercises will be provided in near future in an Answers Manual. This text will appeal to advanced undergraduate and graduate students, as well as researchers and practitioners in engineering, medical sciences, business, social sciences or agriculture. The material discussed in this book is enough for undergraduate and graduate courses. It consists of 5 chapters. Chapter 1 is devoted to the basic concept of probability. Chapters 2 and 3 deal with the concept of a random variable and its distribution and related topics. Chapters 4 and 5 presents an overview of statistical inference, discuss the standard topics of parametric statistical inference, namely, point estimation, interval estimation and testing hypotheses.