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Mathematical Statistics: Basic Ideas and Selected Topics, Vol I (2nd Edition)
by Peter J. Bickel, Kjell A. Doksum
Prentice Hall | 2000-09-27, 2nd edition | ISBN-10: 013850363X
*大小：30.3MB。前言等不算正文的部分和正文加在一起共574页。
*格式：PDF。
内容简介：
This classic, time-honored introduction to the theory and practice of statistics modeling and inference reflects the changing focus of contemporary Statistics. Coverage begins with the more general nonparametric point of view and then looks at parametric models as submodels of the nonparametric ones which can be described smoothly by Euclidean parameters. Although some computational issues are discussed, this is very much a book on theory. It relates theory to conceptual and technical issues encountered in practice, viewing theory as suggestive for practice, not prescriptive. It shows readers how assumptions which lead to neat theory may be unrealistic in practice. Statistical Models, Goals, and Performance Criteria. Methods of Estimation. Measures of Performance, Notions of Optimality, and Construction of Optimal Procedures in Simple Situations. Testing Statistical Hypotheses: Basic Theory. Asymptotic Approximations. Multiparameter Estimation, Testing and Confidence Regions. A Review of Basic Probability Theory. More Advanced Topics in Analysis and Probability. Matrix Algebra. For anyone interested in mathematical statistics working in statistics, bio-statistics, economics, computer science, and mathematics.
*目录：
CONTENTS
PREFACE TO THE SECOND EDITION: VOLUME I xiii
PREFACE TO THE FIRST EDITION xvii
I STATISTICAL MODELS, GOALS, AND PERFORMANCE CRITERIA 1
1.1 Data, Models, Parameters, and Statistics 1
1.1.1 Data and Models I
1.1.2 Parametrization and Parameters 6
1.1.3 Statistics as Functions on the Sample Space 8
1.1.4 Examples, Regression Models 9
1.2 Bayesian Models 12
1.3 The Decision Theoretic Framework 16
1.3.1 Components of the Decision Theory Framework 17
1.3.2 Comparison of Decision Procedures 24
1.3.3 Bayes and Minimax Criteria 26
1.4 Prediction 32
1.5 Sufficiency 41
1.6 Exponential Families 49
1.6.1 The One-Parameter Case 49
1.6.2 The Multiparameter Case 53
1.6.3 Building Exponential Families 56
1.6.4 Properties of Exponential Families 58
1.6.5 Conjugate Families of Prior Distributions 62
1.7 Problems and Complements 66
1.8 Notes 95
1.9 References 96
2 METHODS OF ESTIMATION 99
2.1 Basic Heuristics of Estimation 99
2.1.1 Minimum Contrast Estimates; Estimating Equations 99
2.1.2 The Plug-In and Extension Principles 102
2.2 Minimum Contrast Estimates and Estimating Equations 107
2.2.1 Least Squares and Weighted Least Squares 107
2.2.2 Maximum Likelihood 114
2.3 Maximum Likelihood in Multiparameter Exponential Families 121
*2.4 Algorithmic Issues 127
2.4.1 The Method of Bisection 127
2.4.2 Coordinate Ascent 129
2.4.3 The Newton-Raphson Algorithm 132
2.4.4 The EM (Expectation/Maximization) Algorithm 133
2.5 Problems and Complements 138
2.6 Notes 158
2.7 References 159
3 MEASURES OF PERFORMANCE 161
3.1 Introduction 161
3.2 Bayes Procedures 161
3.3 Minimax Procedures 170
*3.4 Unbiased Estimation and Risk Inequalities 176
3.4.1 Unbiased Estimation, Survey Sampling 176
3.4.2 The Information Inequality 179
*3.5 Nondecision Theoretic Criteria 188
3.5.1 Computation 188
3.5.2 Interpretability 189
3.5.3 Robustness 190
3.6 Problems and Complements 197
3.7 Notes 210
3.8 References 211
4 TESTING AND CONFIDENCE REGIONS 213
4.1 Introduction 213
4.2 Choosing a Test Statistic: The Neyman-Pearson Lemma 223
4.3 Uniformly Most Powerful Tests and Monotone Likelihood Ratio
Models 227
4.4 Confidence Bounds, Intervals, and Regions 233
4.5 The Duality Between Confidence Regions and Tests 241
*4.6 Uniformly Most Accurate Confidence Bounds 248
*4.7 Frequentist and Bayesian Formulations 251
4.8 Prediction Intervals 252
4.9 Likelihood Ratio Procedures 255
4.9.1 Introduction 255
4.9.2 Tests for the Mean of a Normal Distribution-Matched Pair
Experiments 257
4.9.3 Tests and Confidence Intervals for the Difference in Means of
Two Normal Populations 261
4.9.4 The Two-Sample Problem with Unequal Variances 264
4.9.5 Likelihood Ratio Procedures for Bivariate Normal
Distributions 266
4.10 Problems and Complements 269
4.11 Notes 295
4.12 References 295
5 ASYMPTOTIC APPROXIMATIONS 297
5.1 Introduction: The Meaning and Uses of Asymptotics 297
5.2 Consistency 301
5.2.1 Plug-In Estimates and MLEs in Exponential Family Models 301
5.2.2 Consistency of Minimum Contrast Estimates 304
5.3 First- and Higher-Order Asymptotics: The Delta Method with
Applications 306
5.3.1 The Delta Method for Moments 306
5.3.2 The Delta Method for In Law Approximations 311
5.3.3 Asymptotic Normality of the Maximum Likelihood Estimate
in Exponential Families 322
5.4 Asymptotic Theory in One Dimension 324
5.4.1 Estimation: The Multinomial Case 324
*5.4.2 Asymptotic Normality of Minimum Contrast and M-Estimates 327
*5.4.3 Asymptotic Normality and Efficiency of the MLE 331
*5.4.4 Testing 332
*5.4.5 Confidence Bounds 336
5.5 Asymptotic Behavior and Optimality of the Posterior Distribution 337
5.6 Problems and Complements 345
5.7 Notes 362
5.8 References 363
6 INFERENCE IN THE MULTIPARAMETER CASE 365
6.1 Inference for Gaussian Linear Models 365
6.1.1 The Classical Gaussian Linear Model 366
6.1.2 Estimation 369
6.1.3 Tests and Confidence Intervals 374
*6.2 Asymptotic Estimation Theory in p Dimensions 383
6.2.1 Estimating Equations 384
6.2.2 Asymptotic Normality and Efficiency of the MLE 386
6.2.3 The Posterior Distribution in the Multiparameter Case 391
*6.3 Large Sample Tests and Confidence Regions 392
6.3.1 Asymptotic Approximation to the Distribution of the
Likelihood Ratio Statistic 392
6.3.2 Wald's and Rao's Large Sample Tests 398
*6.4 Large Sample Methods for Discrete Data 400
6.4.1 Goodness-of-Fit in a Multinomial Model. Pearson's X2
Test 401
6.4.2 Goodness-of-Fit to Composite Multinomial Models.
Contingency Tables 403
6.4.3 Logistic Regression for Binary Responses 408
*6.5 Generalized Linear Models 411
*6.6 Robustness Properties and Semiparametric Models 417
6.7 Problems and Complements 422
6.8 Notes 438
6.9 References 438
A A REVIEW OF BASIC PROBABILITY THEORY 441
A.I The Basic Model 441
A.2 Elementary Properties of Probability Models 443
A.3 Discrete Probability Models 443
A.4 Conditional Probability and Independence 444
A.5 Compound Experiments 446
A.6 Bernoulli and Multinomial Trials, Sampling With and Without
Replacement 447
A.7 Probabilities On Euclidean Space 448
A.8 Random Variables and Vectors: Transformations 451
A.9 Independence of Random Variables and Vectors 453
A.IO The Expectation of a Random Variable 454
A.II Moments 456
A.12 Moment and Cumulant Generating Functions 459
A. t3 Some Classical Discrete and Continuous Distributions
A.14 Modes of Convergence of Random Variables and Limit Theorems
A. I5 Further Limit Theorems and Inequalities
A.16 Poisson Process
A.17 Notes
A.18 References
B ADDITIONAL TOPICS IN PROBABILITY AND ANALYSIS
B.I Conditioning by a Random Variable or Vector
B.I.l The Discrete Case
B.I.2 Conditional Expectation for Discrete Variables
B.1.3 Properties of Conditional Expected Values
B.IA Continuous Variables
B.l.5 Comments on the General Case
B.2 Distribution Theory for Transformations of Random Vectors
B.2.1 The Basic Framework
B.2.2 The Gamma and Beta Distributions
B.3 Distribution Theory for Samples from a Normal Population
B.3.1 The X2
, F, and t Distributions
B.3.2 Orthogonal Transformations
BA The Bivariate Normal Distribution
B.5 Moments of Random Vectors and Matrices
B.5.1 Basic Properties of Expectations
B.5.2 Properties of Variance
B.6 The Multivariate Normal Distribution
B.6.1 Definition and Density
B.6.2 Basic Properties. Conditional Distributions
B.7 Convergence for Random Vectors: Op and Op Notation
B.8 Multivariate Calculus
B.9 Convexity and Inequalities
B.1O Topics in Matrix Theory and Elementary Hilbert Space Theory
B.1O.1 Symmetric Matrices
B,10.2 Order on Symmetric Matrices
B.10.3 Elementary Hilbert Space Theory
B.Il Problems and Complements
B.12 Notes
B.13 References
•
XI
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C TABLES
Table I The Standard Nonna! Distribution
Table I' Auxiliary Table of the Standard Normal Distribution
Table II t Distribution Critical Values
Table In x2
Distribution Critical Values
Table IV F Distribution Critical Values
INDEX
541
542
543
544
545
546
547
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