1st Edition

Expansions and Asymptotics for Statistics

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

    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.

    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

    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

    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

    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




    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