1st Edition

# Expansions and Asymptotics for Statistics

By Christopher G. Small Copyright 2010
357 Pages 28 B/W Illustrations
by Chapman & Hall

357 Pages
by Chapman & Hall

Also available as eBook on:

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.

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 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 principle of stationary phase
Towards the Barndorff–Nielsen formula
Ratios of sums of random variables
Distributions of M-estimators
The Edgeworth expansion
Mean, median and mode
The method of Darboux
Applications to common distributions

Summation of Series
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

### Biography

Christopher G. Small is a professor in the Department of Statistics and Actuarial Science at the University of Waterloo in Ontario, Canada.

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