Small C.G. Expansions and Asymptotics for Statistics

Chapman and Hall/CRC, 2010. – 338 p. – ISBN: 1584885904, 9781584885900Asymptotic 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.
About the Author
Christopher G. Small is a professor in the Department of Statistics and Actuarial Science at the University of Waterloo in Ontario, Canada.Expansions and approximations.
The role of asymptotics.
Mathematical preliminaries.
Two complementary approaches.
Problems.
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.
Problems.
Pad´e approximants and continued fractions.
The Pad´e table.
Pad´e 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.
Problems.
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.
Problems.
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.
Problems.
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.
Problems.
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.
Problems.
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.
Problems.
Glossary of symbols.
Useful limits, series and products.