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Springer Texts in Statistics Advisors: George Casella Stephen Fienberg Ingram Olkin

Springer Texts in Statistics Alfred: Elements of Statistics for the Life and Social Sciences Berger: An Introduction to Probability and Stochastic Processes Bilodeau and Brenner: Theory of Multivariate Statistics Blom: Probability and Statistics: Theory and Applications Brockwell and Davis: Introduction to Times Series and Forecasting, Second Edition Chow and Teicher: Probability Theory: Independence, Interchangeability, Martingales, Third Edition Christensen: Advanced Linear Modeling: Multivariate, Time Series, and Spatial Data: Nonparametric Regression and Response Surface Maximization, Second Edition Christensen: Log-Linear Models and Logistic Regression, Second Edition Christensen: Plane Answers to Complex Questions: The Theory of Linear Models, Third Edition Creighton: A First Course in Probability Models and Statistical Inference Davis: Statistical Methods for the Analysis of Repeated Measurements Dean and Voss: Design and Analysis of Experiments du Toit, Steyn, and Stumpf: Graphical Exploratory Data Analysis Durrett: Essentials of Stochastic Processes Edwards: Introduction to Graphical Modelling, Second Edition Finkelstein and Levin: Statistics for Lawyers Flury: A First Course in Multivariate Statistics Jobson: Applied Multivariate Data Analysis, Volume I: Regression and Jobson: Applied Multivariate Data Analysis, Volume II: Categorical and Experimental Design Multivariate Methods Edition Kalbfleisch: Probability and Statistical Inference, Volume I: Probability, Second Kalbfleisch: Probability and Statistical Inference, Volume II: Statistical Inference, Second Edition Karr: Probability Keyfitz: Applied Mathematical Demography, Second Edition Kiefer: Introduction to Statistical Inference Kokoska and Nevison: Statistical Tables and Formulae Kulkarni: Modeling, Analysis, Design, and Control of Stochastic Systems Lange: Applied Probability Lehmann: Elements of Large-Sample Theory Lehmann: Testing Statistical Hypotheses, Second Edition Lehmann and Casella: Theory of Point Estimation, Second Edition Lindman: Analysis of Variance in Experimental Design Lindsey: Applying Generalized Linear Models (continued after index)

Jun Shao Mathematical Statistics Second Edition

Jun Shao Department of Statistics University of Wisconsin, Madison Madison, WI 53706-1685 USA shao@stat.wisc.edu Editorial Board George Casella Department of Statistics University of Florida Gainesville, FL 32611-8545 USA With 7 figures. Stephen Fienberg Department of Statistics Carnegie Mellon University Stanford University Pittsburgh, PA 15213-3890 Stanford, CA 94305 USA USA Ingram Olkin Department of Statistics Library of Congress Cataloging-in-Publication Data Shao, Jun. Mathematical statistics / Jun Shao.—2nd ed. p. cm.— (Springer texts in statistics) Includes bibliographical references and index. ISBN 0-387-95382-5 (alk. paper) 1. Mathematical statistics. QA276.S458 519.5—dc21 I. Title. 2003 2003045446 II. Series. ISBN 0-387-95382-5 Printed on acid-free paper. ISBN-13 978-0-387-95382-3 © 2003 Springer Science+Business Media, LLC. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC., 233 Spring St., New York, N.Y., 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Use in connection with any form of Printed in the United States of America. 9 8 7 6 5 4(corrected printing as of 4 th printing, 2007) springer.com

To Guang, Jason, and Annie

Preface to the First Edition This book is intended for a course entitled Mathematical Statistics oﬀered at the Department of Statistics, University of Wisconsin-Madison. This course, taught in a mathematically rigorous fashion, covers essential ma- terials in statistical theory that a ﬁrst or second year graduate student typically needs to learn as preparation for work on a Ph.D. degree in statis- tics. The course is designed for two 15-week semesters, with three lecture hours and two discussion hours in each week. Students in this course are assumed to have a good knowledge of advanced calculus. A course in real analysis or measure theory prior to this course is often recommended. Chapter 1 provides a quick overview of important concepts and results in measure-theoretic probability theory that are used as tools in math- ematical statistics. Chapter 2 introduces some fundamental concepts in statistics, including statistical models, the principle of suﬃciency in data reduction, and two statistical approaches adopted throughout the book: statistical decision theory and statistical inference. Each of Chapters 3 through 7 provides a detailed study of an important topic in statistical de- cision theory and inference: Chapter 3 introduces the theory of unbiased estimation; Chapter 4 studies theory and methods in point estimation un- der parametric models; Chapter 5 covers point estimation in nonparametric settings; Chapter 6 focuses on hypothesis testing; and Chapter 7 discusses interval estimation and conﬁdence sets. The classical frequentist approach is adopted in this book, although the Bayesian approach is also introduced (§2.3.2, §4.1, §6.4.4, and §7.1.3). Asymptotic (large sample) theory, a cru- cial part of statistical inference, is studied throughout the book, rather than in a separate chapter. About 85% of the book covers classical results in statistical theory that are typically found in textbooks of a similar level. These materials are in the Statistics Department’s Ph.D. qualifying examination syllabus. This part of the book is inﬂuenced by several standard textbooks, such as Casella and vii

viii Preface to the First Edition Berger (1990), Ferguson (1967), Lehmann (1983, 1986), and Rohatgi (1976). The other 15% of the book covers some topics in modern statistical theory that have been developed in recent years, including robustness of the least squares estimators, Markov chain Monte Carlo, generalized linear models, quasi-likelihoods, empirical likelihoods, statistical functionals, generalized estimation equations, the jackknife, and the bootstrap. In addition to the presentation of fruitful ideas and results, this book emphasizes the use of important tools in establishing theoretical results. Thus, most proofs of theorems, propositions, and lemmas are provided or left as exercises. Some proofs of theorems are omitted (especially in Chapter 1), because the proofs are lengthy or beyond the scope of the book (references are always provided). Each chapter contains a number of examples. Some of them are designed as materials covered in the discussion section of this course, which is typically taught by a teaching assistant (a senior graduate student). The exercises in each chapter form an important part of the book. They provide not only practice problems for students, but also many additional results as complementary materials to the main text. The book is essentially based on (1) my class notes taken in 1983-84 when I was a student in this course, (2) the notes I used when I was a teaching assistant for this course in 1984-85, and (3) the lecture notes I prepared during 1997-98 as the instructor of this course. I would like to express my thanks to Dennis Cox, who taught this course when I was a student and a teaching assistant, and undoubtedly has inﬂuenced my teaching style and textbook for this course. I am also very grateful to students in my class who provided helpful comments; to Mr. Yonghee Lee, who helped me to prepare all the ﬁgures in this book; to the Springer-Verlag production and copy editors, who helped to improve the presentation; and to my family members, who provided support during the writing of this book. Madison, Wisconsin January 1999 Jun Shao

Preface to the Second Edition 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 semipara- metric 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 con- cepts 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 L´evy-Cram´er 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, propo- sitions, 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 notation and a list of abbreviations, which are appendices of the original book, are given after the references. ix

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