# Introduction to Statistical Limit Theory

#### By **Alan M. Polansky**

Chapman and Hall/CRC – 2011 – 645 pages

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.

**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 r^{th} 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.*

**Alan M. Polansky** is an associate professor in the Division of Statistics at Northern Illinois University. Dr. Polansky is the author of *Observed Confidence Levels: Theory and Application* (CRC Press, October 2007). His research interests encompass nonparametric statistics and industrial applications of statistics.

Name: Introduction to Statistical Limit Theory (Hardback) – Chapman and Hall/CRC

Description: By Alan M. Polansky. 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...

Categories: Statistical Theory & Methods, Statistical Computing, Statistics for the Biological Sciences