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More About This Textbook
Overview
Asymptotic methods provide important tools for approximating and analysing functions that arise in probability and statistics. Moreover, the conclusions of asymptotic analysis often supplement the conclusions obtained by numerical methods. Providing a broad toolkit of analytical methods, Expansions and Asymptotics for Statistics shows how asymptotics, when coupled with numerical methods, becomes a powerful way to acquire a deeper understanding of the techniques used in probability and statistics.
The book first discusses the role of expansions and asymptotics in statistics, the basic properties of power series and asymptotic series, and the study of rational approximations to functions. With a focus on asymptotic normality and asymptotic efficiency of standard estimators, it covers various applications, such as the use of the delta method for bias reduction, variance stabilisation, and the construction of normalising transformations, as well as the standard theory derived from the work of R.A. Fisher, H. Cramér, L. Le Cam, and others. The book then examines the close connection between saddle-point approximation and the Laplace method. The final chapter explores series convergence and the acceleration of that convergence.
What People Are Saying
From the Publisher
This book will be an excellent resource for researchers and graduate students who need a deeper understanding of functions arising in probability and statistics than that provided by numerical techniques.—Eduardo Gutiérrez-Peña, International Statistical Review, 2012
This outstanding book is rich in contents and excellent in readability. … I enjoyed reading this book and found this book valuable in my research as well as in my understanding of expansions and asymptotics as they arise often in statistics. The author has to be commended for his contribution to our profession in getting this book out.
—Subir Ghosh, Technometrics, May 2012
I have found this book very useful not only for the specialists in asymptotics but especially for all those who wish to learn more from this field and to see the inter-relations between different approaches.
—Jaromir Antoch, Zentralblatt MATH
This is an excellent book for researchers interested in asymptotics, especially those working on (mathematical) statistics or applied probability. … The book contains a compilation of different techniques to deal with series expansions and approximations with statistical applications. Examples are focused on the approximation of probability densities, distributions and likelihoods.
—Javier Carcamo, Mathematical Reviews
Product Details
Meet the Author
Christopher G. Small is a professor in the Department of Statistics and Actuarial Science at the University of Waterloo in Ontario, Canada.
Table of Contents
Introduction
Expansions and approximations The role of asymptotics Mathematical preliminaries Two complementary approaches
General Series Methods
A quick overview Power series Enveloping series Asymptotic series Superasymptotic and hyperasymptotic series Asymptotic series for large samples Generalised asymptotic expansions Notes
Padé Approximants and Continued Fractions
The Padé table Padé approximations for the exponential function Two applications Continued fraction expansions A continued fraction for the normal distribution Approximating transforms and other integrals Multivariate extensions Notes
The Delta Method and Its Extensions
Introduction to the delta method Preliminary results The delta method for moments Using the delta method in Maple Asymptotic bias Variance stabilising transformations Normalising transformations Parameter transformations Functions of several variables Ratios of averages The delta method for distributions The von Mises calculus Obstacles and opportunities: robustness
Optimality and Likelihood Asymptotics
Historical overview The organisation of this chapter The likelihood function and its properties Consistency of maximum likelihood Asymptotic normality of maximum likelihood Asymptotic comparison of estimators Local asymptotics Local asymptotic normality Local asymptotic minimaxity Various extensions
The Laplace Approximation and Series
A simple example The basic approximation The Stirling series for factorials Laplace expansions in Maple Asymptotic bias of the median Recurrence properties of random walks Proofs of the main propositions Integrals with the maximum on the boundary Integrals of higher dimension Integrals with product integrands Applications to statistical inference Estimating location parameters Asymptotic analysis of Bayes estimators Notes
The Saddle-Point Method
The principle of stationary phase Perron’s saddle-point method Harmonic functions and saddle-point geometry Daniels’saddle-point approximation Towards the Barndorff–Nielsen formula Saddle-point method for distribution functions Saddle-point method for discrete variables Ratios of sums of random variables Distributions of M-estimators The Edgeworth expansion Mean, median and mode Hayman’s saddle-point approximation The method of Darboux Applications to common distributions
Summation of Series
Advanced tests for series convergence Convergence of random series Applications in probability and statistics Euler–Maclaurin sum formula Applications of the Euler–Maclaurin formula Accelerating series convergence Applications of acceleration methods Comparing acceleration techniques Divergent series
Glossary of Symbols
Useful Limits, Series and Products
References
Index