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A course in mathematical statistics and large sample theory
Bhattacharya R., Lin L., Patrangenaru V., Springer International Publishing, New York, NY, 2016. 389 pp. Type: Book (978-1-493940-30-1)
Date Reviewed: Mar 28 2017

Researchers, mathematicians, and statisticians at the University of Arizona, the University of Notre Dame, and Florida State University are the authors of this book. It is clearly the result of the long-term collection of rich educational material containing both traditional and recent topics. It deals with advanced statistical theory with a special focus on statistical inference and large sample theory, aiming to cover the material for a modern two-semester graduate course in mathematical statistics. Although the area of mathematical statistics has a very long history in publication and many fine books exist, a new textbook that brings a fresh point of view to a comprehensive graduate course on the subject is always welcome. In this era of big data and data science, where researchers are focused on fast and efficient, usually black-box algorithms, such authoring efforts, aiming to promote the role of theoretical statistics in data analysis and inference, are praiseworthy.

The main characteristic and strength of the book is the wide range of subjects covered. Of course, some of these topics are not developed to a great extent, but the reader eventually gains global and comprehensive knowledge of concepts and methods, basic for further reading or even research. Furthermore, the context is presented gradually with a reasonable flow, as I describe below. However, the study of the book will not be easy for all readers and requires strong mathematical and statistical background, especially because several subjects contain notions from measure theoretic probability.

The material consists of 15 chapters organized in three parts. Part 1 is devoted to basic parametric mathematical statistics. Part 2 deals with more advanced topics related to large sample theory, and Part 3 describes quite briefly special topics related to the preceding theory. Each chapter ends with a number of exercises and occasionally with projects for students, depicting the educational aims of the book. Solutions to selected exercises are provided at the end of the book. Sections of notes and suggested references for further study are also provided in each chapter. Furthermore, theoretical parts are often accompanied by several examples.

Part 1 opens with an introductory chapter describing the various aspects of statistical inference and introducing two of the most known sampling methods: simple random and stratified random sampling. In the chapter 2, statistical inference is connected with decision theory. Specifically, basic notions such as loss functions, risk functions, decision rules, and admissibility are discussed, with respect to the observations from a sample. Because the definition of these notions is based on unknown population parameters, the next chapter describes methods for estimating them. The maximum likelihood method and the method of moments are described, along with a special and detailed focus on Bayesian rules and estimators. Because optimal statistical inference is based on appropriate statistics computed from samples, chapter 4 continues with a desirable property of statistics: sufficiency. The notion of a sufficient statistic is especially studied in exponential parametric families of distributions. The final (and largest) chapter of the first part is devoted to testing hypotheses and specifically to the theory of optimal parametric tests, covering several theoretical aspects and having as a starting point the simple hypotheses and the theory developed by Neyman and Pearson.

Part 2 opens with the fundamentals of large sample theory. Various notions related to convergence are discussed in chapter 6, together with an introduction to consistent estimators and their asymptotic distributions. Chapter 7 focuses more on various types of estimators and their asymptotic behavior, and chapter 8 presents the asymptotic theory of tests in selected parametric and nonparametric models. Within the general large sample theory, the nonparametric bootstrap methodology for confidence intervals is briefly presented (chapter 9). The last chapter of this part describes the basics of nonparametric curve estimation, specifically for the density function and for regression functions.

Part 3 contains special topics related to statistical inference and large sample theory. In chapters 11 to 15, advanced subjects like asymptotic expansions for distributions, nonparametric inference on non-Euclidean spaces, simultaneous testing of multiple null hypotheses, Markov chain Monte Carlo simulation, machine learning classification models, principal components analysis, and sequential analysis are presented.

Finally, there are four appendices that are useful as references when studying the previous chapters. Appendix A describes and provides useful properties of key discrete and continuous distributions; Appendix B reviews moment-generating functions; Appendix C deals with the computation of power of selected optimal tests; and Appendix D provides central limit theorems.

Overall, the book is very advanced and is recommended to graduate students with sound statistical backgrounds, as well as to teachers, researchers, and practitioners who wish to acquire more knowledge on mathematical statistics and large sample theory.

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Reviewer:  Lefteris Angelis Review #: CR145147 (1706-0343)
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