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Mathematical Statistics PDF

607 Pages·2003·7.723 MB·English
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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 Experimental Design Jobson: Applied Multivariate Data Analysis, Volume II: Categorical and Multivariate Methods Kalbfleisch: Probability and Statistical Inference, Volume I: Probability, Second Edition 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 [email protected] Editorial Board George Casella Stephen Fienberg Ingram Olkin Department of Statistics Department of Statistics Department of Statistics University of Florida Carnegie Mellon University Stanford University Gainesville,FL 32611-8545 Pittsburgh,PA 15213-3890 Stanford,CA 94305 USA USA USA With 7 figures. Library ofCongress 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. I.Title. II.Series. QA276.S458 2003 519.5—dc21 2003045446 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. Use in connection with any form of 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. Printed in the United States ofAmerica. th 9 8 7 6 5 4(corrected printing as of 4 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 typicallyneedstolearnaspreparationforworkonaPh.D.degreeinstatis- 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 through7 providesa detailed study of an important topic in statisticalde- cision theory and inference: Chapter 3 introduces the theory of unbiased estimation; Chapter 4 studies theory and methods in point estimation un- derparametricmodels;Chapter5coverspointestimationinnonparametric settings; Chapter 6 focuses on hypothesis testing; and Chapter 7 discusses interval estimation and confidence sets. The classical frequentist approach is adoptedin this book, althoughthe Bayesianapproachis alsointroduced ( 2.3.2, 4.1, 6.4.4, and 7.1.3). Asymptotic (large sample) theory, a cru- § § § § cialpartofstatisticalinference,isstudiedthroughoutthebook,ratherthan in a separate chapter. About 85%ofthe book coversclassicalresults instatisticaltheory that aretypicallyfoundintextbooksofasimilarlevel. Thesematerialsareinthe Statistics Department’s Ph.D. qualifying examination syllabus. This part ofthebookisinfluencedbyseveralstandardtextbooks,suchasCasellaand vii viii Preface to the First Edition Berger(1990),Ferguson(1967),Lehmann(1983,1986),andRohatgi(1976). The other 15%of the book coverssome 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. Someofthemaredesignedasmaterialscoveredinthediscussion section of this course, which is typically taught by a teaching assistant (a senior graduatestudent). The exercisesin eachchapter formanimportant 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, whohelpedmetoprepareallthefiguresinthisbook;totheSpringer-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 Jun Shao January 1999 Preface to the Second Edition Inadditiontocorrectingtyposanderrorsandmakingabetterpresentation, 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,twonew sectionsarecreatedtointroduce 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 treatedin 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- ceptsofweakconvergenceandtightnessareintroduced;proofstosomekey results in asymptotic theory, such as the dominated convergence theorem and monotone convergencetheorem, the L´evy-Cram´ercontinuity 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 originalbook, indices on definitions, examples, theorems, propo- sitions, corollaries, and lemmas are included in the subject index. In the newedition,theyareinaseparateindexgivenintheendofthebook(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 x Preface to the Second Edition The most significant change in notation is the notation for a vector. In the text of the new edition, a k-dimensional vector is denoted by c = (c ,...,c ), whether it is treated as a column or a row vector (which is not 1 k important if matrix algebra is not considered). When matrix algebra is involved, any vector c is treated as a k 1 matrix (a column vector) and × its transpose cτ is treated as a 1 k matrix (a row vector). Thus, for × c=(c ,...,c ),cτc=c2+ +c2 andccτ isthe k k matrixwhose(i,j)th 1 k 1 ··· k × element is c c . i j I would like to thank reviewers of this book for their constructive com- ments, the Springer-Verlag production and copy editors, students in my classes, and two teaching assistants, Mr. Bin Cheng and Dr. Hansheng Wang, who provided help in preparing the new edition. Any remaining errors are of course my own responsibility, and a correction of them may be found on my web page http://www.stat.wisc.edu/˜shao. Madison, Wisconsin Jun Shao April, 2003

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