Mathematical statistics. 2nd ed.

*(English)*Zbl 1018.62001
Springer Texts in Statistics. New York, NY: Springer. xvi, 591 p. (2003).

[For the review of the first edition from 1999 see Zbl 0935.62004.]

From the preface: In addition to correcting typos and errors and making a better presentation, the main effort in preparing this new edition is adding some new material to Chapter 1 (Probability Theory) and a number of new exercises to each chapter. Furthermore, two new sections are created to introduce semiparametric models and methods (§5.1.4) and to study the asymptotic accuracy of confidence sets (§7.3.4). The structure of the book remains the same.

In Chapter 1 of the new edition, moment generating and characteristic functions are treated in more detail and a proof of the uniqueness theorem is provided; some useful moment inequalities are introduced; discussions on conditional independence, Markov chains, and martingales are added, as a continuation of the discussion of conditional expectations; the concepts of weak convergence and tightness are introduced; proofs to some key results in asymptotic theory, such as the dominated convergence theorem and monotone convergence theorem, the Lévy-Cramér continuity theorem, the strong and weak laws of large numbers, and Lindeberg’s central limit theorem, are included; and a new section (§1.5.6) is created to introduce Edgeworth and Cornish-Fisher expansions. As a result, Chapter 1 of the new edition is self-contained for important concepts, results, and proofs in probability theory with emphasis in statistical applications.

Since the original book was published in 1999, I have been using it as a textbook for a two-semester course in mathematical statistics. Exercise problems accumulated during my teaching are added to this new edition. Some exercises that are too trivial have been removed.

In the original book, indices on definitions, examples, theorems, propositions, corollaries, and lemmas are included in the subject index. In the new edition, they are in a separate index given in the end of the book (prior to the author index). A list of notations and a list of abbreviations, which are appendices of the original book, are given after the references.

From the preface: In addition to correcting typos and errors and making a better presentation, the main effort in preparing this new edition is adding some new material to Chapter 1 (Probability Theory) and a number of new exercises to each chapter. Furthermore, two new sections are created to introduce semiparametric models and methods (§5.1.4) and to study the asymptotic accuracy of confidence sets (§7.3.4). The structure of the book remains the same.

In Chapter 1 of the new edition, moment generating and characteristic functions are treated in more detail and a proof of the uniqueness theorem is provided; some useful moment inequalities are introduced; discussions on conditional independence, Markov chains, and martingales are added, as a continuation of the discussion of conditional expectations; the concepts of weak convergence and tightness are introduced; proofs to some key results in asymptotic theory, such as the dominated convergence theorem and monotone convergence theorem, the Lévy-Cramér continuity theorem, the strong and weak laws of large numbers, and Lindeberg’s central limit theorem, are included; and a new section (§1.5.6) is created to introduce Edgeworth and Cornish-Fisher expansions. As a result, Chapter 1 of the new edition is self-contained for important concepts, results, and proofs in probability theory with emphasis in statistical applications.

Since the original book was published in 1999, I have been using it as a textbook for a two-semester course in mathematical statistics. Exercise problems accumulated during my teaching are added to this new edition. Some exercises that are too trivial have been removed.

In the original book, indices on definitions, examples, theorems, propositions, corollaries, and lemmas are included in the subject index. In the new edition, they are in a separate index given in the end of the book (prior to the author index). A list of notations and a list of abbreviations, which are appendices of the original book, are given after the references.

Reviewer: Reviewer (Berlin)

##### MSC:

62-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |