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.