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 Wahrscheinlichkeitsrechnung
• Mathematik | Informatik Mathematik Stochastik Mathematische Statistik

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.