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1、Asymptotic Statistics This book is an introduction to the field of asymptotic statistics.The treatment is both practical and mathematically rigorous.In addition to most of the standard topics of an asymptotics course,including likelihood inference,M-estimation,asymptotic efficiency,U-statistics,and
2、rank procedures,the book also presents recent research topics such as sernipararnetric models,the bootstrap,and empirical processes and their applications.One of the unifying themes is the approximation by limit experiments.This entails mainly the local approximation of the classical i.i.d.set-up wi
3、th smooth parameters by location experiments involving a single,normally distributed observation.Thus,even the standard subjects of asymptotic statistics are presented in a novel way.Suitable as a text for a graduate or Masters level statistics course,this book also gives researchers in statistics,p
4、robability,and their applications an overview of the latest research in asymptotic statistics.A.W.van der Vaart is Professor of Statistics in the Department of Mathematics and Computer Science at the Vrije Universiteit,Amsterdam.CAMBRIDGE SERIES IN STATISTICAL AND PROBABILISTIC MATHEMATICS Editorial
5、 Board:R.Gill,Department of Mathematics,Utrecht University B.D.Ripley,Department of Statistics,University of Oxford S.Ross,Department of Industrial Engineering,University of California,Berkeley M.Stein,Department of Statistics,University of Chicago D.Williams,School of Mathematical Sciences,Universi
6、ty of Bath This series of high-quality upper-division textbooks and expository monographs covers all aspects of stochastic applicable mathematics.The topics range from pure and applied statistics to probability theory,operations research,optimization,and mathematical programming.The books contain cl
7、ear presentations of new developments in the field and also of the state of the art in classical methods.While emphasizing rigorous treatment of theoretical methods,the books also contain applications and discussions of new techniques made possible by advances in computational practice.Already publi
8、shed 1.Bootstrap Methods and Their Application,by A.C.Davison and D.V.Hinkley 2.Markov Chains,by J.Norris Asymptotic Statistics A.W.VANDER VAART CAMBRIDGE UNIVERSITY PRESS PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building,Trumpington Street,Cambridge,United Kingdom CA
9、MBRIDGE UNIVERSITY PRESS The Edinburgh Building,Cambridge CB2 2RU,UK http:/www.cup.cam.ac.uk 40 West 20th Street,New York,NY 10011-4211,USA http:/www.cup.org 10 Stamford Road,Oakleigh,Melbourne 3166,Australia Ruiz de Alarcon 13,28014 Madrid,Spain Cambridge University Press 1998 This book is in copyr
10、ight.Subject to statutory exception and to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the written permission of Cambridge University Press.First published 1998 First paperback edition 2000 Printed in the United States of America Type
11、set in Times Roman 10112.5 pt in LKfff(2 TB A catalog record for this book is available from the British Library Library of Congress Cataloging in Publication data Vaart,A.W.van der Asymtotic statistics I A.W.van der Vaart.p.em.-(Cambridge series in statistical and probablistic mathematics)Includes
12、bibliographical references.1.Mathematical statistics-Asymptotic theory.I.Title.II.Series:cambridge series on statistical and probablistic mathematics.CA276.V22 1998 519.5-dc21 98-15176 ISBN 0 521 49603 9 hardback ISBN 0 521 78450 6 paperback To Maryse and Marianne Contents Preface page xiii Notation
13、 page XV 1.Introduction 1 1.1.Approximate Statistical Procedures 1 1.2.Asymptotic Optimality Theory 2 1.3.Limitations 3 1.4.The Index n 4 2.Stochastic Convergence 5 2.1.Basic Theory 5 2.2.Stochastic o and 0 Symbols 12*2.3.Characteristic Functions 13*2.4.Almost-Sure Representations 17*2.5.Convergence
14、 of Moments 17*2.6.Convergence-Determining Classes 18*2.7.Law of the Iterated Logarithm 19*2.8.Linde berg-Feller Theorem 20*2.9.Convergence in Total Variation 22 Problems 24 3.Delta Method 25 3.1.Basic Result 25 3.2.Variance-Stabilizing Transformations 30*3.3.Higher-Order Expansions 31*3.4.Uniform D
15、elta Method 32*3.5.Moments 33 Problems 34 4.Moment Estimators 35 4.1.Method of Moments 35*4.2.Exponential Families 37 Problems 40 5.M-and Z-Estimators 41 5.1.Introduction 41 5.2.Consistency 44 5.3.Asymptotic Normality 51 Vll Vlll Contents*5.4.Estimated Parameters 60 5.5.Maximum Likelihood Estimators
16、 61*5.6.Classical Conditions 67*5.7.One-Step Estimators 71*5.8.Rates of Convergence 75*5.9.Argmax Theorem 79 Problems 83 6.Contiguity 85 6.1.Likelihood Ratios 85 6.2.Contiguity 87 Problems 91 7.Local Asymptotic Normality 92 7.1.Introduction 92 7.2.Expanding the Likelihood 93 7.3.Convergence to a Nor
17、mal Experiment 97 7.4.Maximum Likelihood 100*7.5.Limit Distributions under Alternatives 103*7.6.Local Asymptotic Normality 103 Problems 106 8.Efficiency of Estimators 108 8.1.Asymptotic Concentration 108 8.2.Relative Efficiency 110 8.3.Lower Bound for Experiments 111 8.4.Estimating Normal Means 112
18、8.5.Convolution Theorem 115 8.6.Almost-Everywhere Convolution Theorem 115*8.7.Local Asymptotic Minimax Theorem 117*8.8.Shrinkage Estimators 119*8.9.Achieving the Bound 120*8.10.Large Deviations 122 Problems 123 9.Limits of Experiments 125 9.1.Introduction 125 9.2.Asymptotic Representation Theorem 12
19、6 9.3.Asymptotic Normality 127 9.4.Uniform Distribution 129 9.5.Pareto Distribution 130 9.6.Asymptotic Mixed Normality 131 9.7.Heuristics 136 Problems 137 10.Bayes Procedures 138 10.1.Introduction 138 10.2.Bernstein-von Mises Theorem 140 Contents ix 10.3.Point Estimators 146*10.4.Consistency 149 Pro
20、blems 152 11.Projections 153 11.1.Projections 153 11.2.Conditional Expectation 155 11.3.Projection onto Sums 157*11.4.Hoeffding Decomposition 157 Problems 160 12.U-Statistics 161 12.1.One-Sample U-Statistics 161 12.2.Two-Sample U-statistics 165*12.3.Degenerate U-Statistics 167 Problems 171 13.Rank,S
21、ign,and Permutation Statistics 173 13.1.Rank Statistics 173 13.2.Signed Rank Statistics 181 13.3.Rank Statistics for Independence 184*13.4.Rank Statistics under Alternatives 184 13.5.Permutation Tests 188*13.6.Rank Central Limit Theorem 190 Problems 190 14.Relative Efficiency of Tests 192 14.1.Asymp
22、totic Power Functions 192 14.2.Consistency 199 14.3.Asymptotic Relative Efficiency 201*14.4.Other Relative Efficiencies 202*14.5.Rescaling Rates 211 Problems 213 15.Efficiency of Tests 215 15.1.Asymptotic Representation Theorem 215 15.2.Testing Normal Means 216 15.3.Local Asymptotic Normality 218 15
23、.4.One-Sample Location 220 15.5.Two-Sample Problems 223 Problems 226 16.Likelihood Ratio Tests 227 16.1.Introduction 227*16.2.Taylor Expansion 229 16.3.Using Local Asymptotic Normality 231 16.4.Asymptotic Power Functions 236 X Contents 16.5.Bartlett Correction 238*16.6.Bahadur Efficiency 238 Problem
24、s 241 17.Chi-Square Tests 242 17.1.Quadratic Forms in Normal Vectors 242 17.2.Pearson Statistic 242 17.3.Estimated Parameters 244 17.4.Testing Independence 247*17.5.Goodness-of-Fit Tests 248*17.6.Asymptotic Efficiency 251 Problems 253 18.Stochastic Convergence in Metric Spaces 255 18.1.Metric and No
25、rmed Spaces 255 18.2.Basic Properties 258 18.3.Bounded Stochastic Processes 260 Problems 263 19.Empirical Processes 265 19.1.Empirical Distribution Functions 265 19.2.Empirical Distributions 269 19.3.Goodness-of-Fit Statistics 277 19.4.Random Functions 279 19.5.Changing Classes 282 19.6.Maximal Ineq
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