Polansky Introduction to Statistical Limit Theory

1. Auflage 2011
ISBN: 978-1-4398-8457-7
Verlag: Taylor & Francis
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 645 Seiten

Reihe: Chapman & Hall/CRC Texts in Statistical Science

Helping students develop a good understanding of asymptotic theory, Introduction to Statistical Limit Theory provides a thorough yet accessible treatment of common modes of convergence and their related tools used in statistics. It also discusses how the results can be applied to several common areas in the field.

The author explains as much of the background material as possible and offers a comprehensive account of the modes of convergence of random variables, distributions, and moments, establishing a firm foundation for the applications that appear later in the book. The text includes detailed proofs that follow a logical progression of the central inferences of each result. It also presents in-depth explanations of the results and identifies important tools and techniques. Through numerous illustrative examples, the book shows how asymptotic theory offers deep insight into statistical problems, such as confidence intervals, hypothesis tests, and estimation.

With an array of exercises and experiments in each chapter, this classroom-tested book gives students the mathematical foundation needed to understand asymptotic theory. It covers the necessary introductory material as well as modern statistical applications, exploring how the underlying mathematical and statistical theories work together.

Autoren/Hrsg.

Polansky, Alan M.

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik

Inhaltsverzeichnis

Sequences of Real Numbers and Functions

Introduction

Sequences of Real Numbers

Sequences of Real Functions

The Taylor Expansion

Asymptotic Expansions

Inversion of Asymptotic Expansions

Random Variables and Characteristic Functions

Introduction

Probability Measures and Random Variables
Some Important Inequalities

Some Limit Theory for Events
Generating and Characteristic Functions

Convergence of Random Variables

Introduction

Convergence in Probability
Stronger Modes of Convergence
Convergence of Random Vectors

Continuous Mapping Theorems

Laws of Large Numbers

The Glivenko–Cantelli Theorem
Sample Moments

Sample Quantiles

Convergence of Distributions
Introduction

Weak Convergence of Random Variables

Weak Convergence of Random Vectors

The Central Limit Theorem

The Accuracy of the Normal Approximation

The Sample Moments

The Sample Quantiles

Convergence of Moments

Convergence in rth Mean

Uniform Integrability

Convergence of Moments

Central Limit Theorems
Introduction

Non-Identically Distributed Random Variables
Triangular Arrays
Transformed Random Variables

Asymptotic Expansions for Distributions
Approximating a Distribution

Edgeworth Expansions

The Cornish–Fisher Expansion

The Smooth Function Model

General Edgeworth and Cornish–Fisher Expansions
Studentized Statistics

Saddlepoint Expansions

Asymptotic Expansions for Random Variables
Approximating Random Variables
Stochastic Order Notation
The Delta Method

The Sample Moments

Differentiable Statistical Functionals
Introduction

Functional Parameters and Statistics

Differentiation of Statistical Functionals

Expansion Theory for Statistical Functionals

Asymptotic Distribution
Parametric Inference

Introduction

Point Estimation

Confidence Intervals

Statistical Hypothesis Tests

Observed Confidence Levels

Bayesian Estimation
Nonparametric Inference

Introduction

Unbiased Estimation and U-Statistics
Linear Rank Statistics
Pitman Asymptotic Relative Efficiency

Density Estimation

The Bootstrap
Appendix A: Useful Theorems and Notation
Appendix B: Using R for Experimentation
References
Exercises and Experiments appear at the end of each chapter.