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Publisher:

Chapman & Hall/CRC

Publication Date:

2011

Number of Pages:

625

Format:

Hardcover

Series:

Texts in Statistical Science

Price:

89.95

ISBN:

9781420076608

Category:

Textbook

[Reviewed by , on ]

Peter Rabinovitch

06/25/2011

*Introduction to Statistical Limit Theory* covers the basic limit theorems of probability and their applications to statistics. It does not presuppose any measure theory, although a decent background in analysis is required (say at the level of Spivak's *Calculus*), and of course enough background in statistics to make the applications meaningful.

This book is clearly aimed at the student learning the material. The prose is clear and graphs are used well to illustrate concepts. Students will really like the large number of worked concrete examples. For example, many probability texts will show that the sum of two uniformly integrable random variables is uniformly integrable. This book will also show examples of several particular families that are uniformly integrable, such as random variables with a Gaussian distribution with mean 0 and variance v(n), 0 < v(n) < ∞.

Each chapter has exercises and R projects that would be very helpful to solidify the understanding of the material in the chapter. These and the worked examples would make this an excellent choice for the student attempting to learn the material through self-study, except for the many errors that may confuse the student reader. For example:

- Ex. 1.11 and Ex. 1.13 discuss uniform convergence of a sequence of functions, but has the sequence and the limiting function confused in some places.
- Theorem 2.6 on page 60 uses an integral with respect to dF, where F is the cumulative distribution function (cdf). However, the cdf is not defined until page 147. And, I did not find anywhere a discussion of integration with respect to dF.
- On page 106, Ex. 3.4 we have (in essence) P[X
- The Kolmogorov distance between distributions is never defined, but is used for example in Figure 4.11.

If a future edition corrects the errors, then this book will certainly be worth considering. Until then, I much prefer *Large Sample Techniques for Statistics* by Jiming Jiang which covers similar territory, but with few (if any) errors.

Peter Rabinovitch is a Systems Architect at Research in Motion and a PhD student in probability. When not working, or trying to finish his thesis, he likes to review math books while drinking iced cappuccinos.

**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

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

Linear Rank Statistics

Pitman Asymptotic Relative Efficiency

Density Estimation

The Bootstrap

**Appendix A: Useful Theorems and NotationAppendix B: Using R for Experimentation**

**References**

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