Small Expansions and Asymptotics for Statistics

1. Auflage 2010
ISBN: 978-1-4200-1102-9
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 357 Seiten

Reihe: Chapman & Hall/CRC Monographs on Statistics & Applied Probability

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.

Zielgruppe

Researchers and graduate students in statistics, mathematics, and econometrics.

Autoren/Hrsg.

Small, Christopher G.

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik Mathematische Statistik
• Mathematik | Informatik Mathematik Stochastik Wahrscheinlichkeitsrechnung

Inhaltsverzeichnis

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

Über Autor(innen)

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