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From Finite Sample to Asymptotic Methods in Statistics

P. K. Sen, J. M. Singer, and A.C. Pedroso de Lima
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Series in Statistical and Probabilistic Mathematics
[Reviewed by
Manan Shah
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From Finite Sample to Asymptotic Methods in Statistics is written as a mid-level graduate statistics text and as a reference for statisticians. It is, in general, difficult to make texts at this level both readable and rigorous, but the authors do just that. What differentiates this text from other advanced texts is that there is a fair amount of written discussion as well as mathematical exposition. This is a necessary but often forgotten stylistic element and is evidence that the authors genuinely had the reader’s interests in mind.

As this is a mid-level graduate statistics text, it is essentially required that the reader have had an introductory measure theory course in the context of probability theory, an introductory graduate course in probability and statistics, a course in linear algebra, and perhaps a course in stochastic processes. The authors do give a fairly involved overview/review of these topics; so that those who are unfamiliar can still become familiar with the necessary definitions and theorems.

As for mathematical readability, this text does an excellent job in keeping the notation consistent, standard, and not cumbersome. From an organizational standpoint, it is a well-structured and logically flowing book; problems appear at the end of each chapter and are numbered in a convenient format: Chapter.Section.ProblemNumber. This avoids the sometimes annoying issue of having to poke around a chapter for problems from a particular section and it also allows one to read through a chapter without an “unnatural” break. There is no “answers to chapter problems” section in this book, but in fairness, texts at this level often do not have such sections.

Chapter 1 is a typical introduction establishing the purpose and scope of the book as well as some motivational examples and some background reading. All readers will find it helpful to go through this chapter since it will familiarize them with some examples that are used later in the book and it will familiarize them with the notation used throughout. While it was mentioned earlier that the notation is standard, there are different standards. As an example: the transpose of a matrix, A, is often written as either AT or At; this text chooses At. As an another example: the kth order statistic can be written as X(k) or X(k) or Xn:k; this text chooses Xn:k.

Chapters 2 and 3 give an overview of estimation theory and hypothesis testing. Most of this material is covered in an introductory graduate course in probability and statistics. Thus these chapters are provided mostly as reference and review, but to also highlight the limits of exact statistical inference and the need for asymptotic methods.

Chapter 4 is a brief overview of statistical decision theory including Bayesian methods. Chapter 5 is a succinct review of stochastic processes, covering discrete- and continuous- time parameter processes with a focus on martingale theory and renewal theory.

Chapter 6 discusses stochastic convergence and probability inequalities. It is not assumed that the reader has necessarily had exposure to measure theory. Those with exposure to measure theory in a probabilistic context will see this chapter more as a review of fundamentals than anything new. Those with exposure to measure theory, but not from the view of probability theory would be advised to carefully read this chapter.

Chapters 7, 8, and 9 are the meat of this book. Chapter 7 gives a discussion on asymptotic distributions, mostly including the following topics: some major central limit theorems, rates of convergence to normality, variance stabilizing transformations, asymptotic distribution of quadratic forms, order statistics, and empirical distributions. Chapter 8 discusses, as the chapter title suggests, the “asymptotic behavior of estimators and tests”. Chapter 9 discusses the goodness-of-fit tests in both the parametric and non-parametric setting with a focus on the Pearsonian GOF test statistic for a multinomial model. All of these chapters are theoretical in nature. The last section of Chapter 8 briefly discusses the bootstrap and jackknife methods as a computational alternative.

Chapters 10 and 11 round out the text. Chapter 10 gives a treatment of regression models and the least-squares methodology. This includes a discussion of linear, non-linear, and non-parametric regression models. Finally, Chapter 11 goes over weak convergence and Gaussian processes. This chapter will require that the reader be familiar with the Wiener process (aka Brownian motion) and the construction of a Brownian bridge. For those unfamiliar with the Wiener process and the resulting theory, the text does not disappoint as the necessary theory is introduced in a detailed and accessible manner. The use of a Brownian bridge is essential in the analysis of the asymptotic distribution of certain test statistics, like the Kolmogorov-Smirnov statistic.

All in all, From Finite Sample to Asymptotic Methods in Statistics, is a very well-written mid-level graduate statistics text.

Manan Shah splits his time: he is a mathematician in industry during the day and an academic researcher in the evenings.

1. Motivation and basic tools; 2. Estimation theory; 3. Hypothesis testing; 4. Elements of statistical decision theory; 5. Stochastic processes: an overview; 6. Stochastic convergence and probability inequalities; 7. Asymptotic distributions; 8. Asymptotic behavior of estimators and tests; 9. Categorical data models; 10. Regression models; 11. Weak convergence and Gaussian processes.