Introduction to Empirical Processes and Semiparametric Inference

Michael R. Kosorok

Springer, Dec 29, 2007 - Mathematics - 483 pages

The goal of this book is to introduce statisticians, and other researchers with a background in mathematical statistics, to empirical processes and semiparametric inference. These powerful research techniques are surpr- ingly useful for studying large sample properties of statistical estimates from realistically complex models as well as for developing new and - proved approaches to statistical inference. This book is more of a textbook than a research monograph, although a number of new results are presented. The level of the book is more - troductory than the seminal work of van der Vaart and Wellner (1996). In fact, another purpose of this work is to help readers prepare for the mathematically advanced van der Vaart and Wellner text, as well as for the semiparametric inference work of Bickel, Klaassen, Ritov and We- ner (1997). These two books, along with Pollard (1990) and Chapters 19 and 25 of van der Vaart (1998), formulate a very complete and successful elucidation of modern empirical process methods. The present book owes much by the way of inspiration, concept, and notation to these previous works.What is perhaps new is the gradual—yetrigorous—anduni?ed way this book introduces the reader to the ?eld.

Contents

• Introduction: 3
• An Overview of Empirical Processes: 9
• 22 Empirical Process Techniques: 13
• 222 Entropy for GlivenkoCantelli and Donsker Theorems: 16
• 223 Bootstrapping Empirical Processes: 19
• 224 The Functional Delta Method: 21
• 225 ZEstimators: 24
• 226 MEstimators: 28
• 23 Other Topics: 30
• 24 Exercises: 32
• 25 Notes: 33
• Overview of Semiparametric Inference: 34
• 32 Score Functions and Estimating Equations: 39
• 33 Maximum Likelihood Estimation: 44
• 34 Other Topics: 47
• 35 Exercises: 48
• Case Studies I: 49
• 41 Linear Regression: 50
• 412 Median Zero Residuals: 52
• 42 Counting Process Regression: 54
• 421 The General Case: 55
• 422 The Cox Model: 59
• 43 The KaplanMeier Estimator: 60
• 44 Efficient Estimating Equations for Regression: 62
• 441 Simple Linear Regression: 66
• 442 A Poisson Mixture Regression Model: 69
• 46 Exercises: 72
• Empirical Processes: 74
• Introduction to Empirical Processes: 75
• Preliminaries for Empirical Processes: 81
• 62 Outer Expectation: 88
• 63 Linear Operators and Functional Differentiation: 93
• 64 Proofs: 96
• 65 Exercises: 100
• 66 Notes: 102
• Stochastic Convergence: 103
• 72 Weak Convergence: 107
• 722 Spaces of Bounded Functions: 113
• 73 Other Modes of Convergence: 115
• 74 Proofs: 120
• 75 Exercises: 125
• 76 Notes: 126
• Empirical Process Methods: 127
• 81 Maximal Inequalities: 128
• 812 Maximal Inequalities for Processes: 131
• 82 The Symmetrization Inequality and Measurability: 138
• 83 GlivenkoCantelli Results: 144
• 84 Donsker Results: 148
• 85 Exercises: 151
• 86 Notes: 153
• Entropy Calculations: 154
• 91 Uniform Entropy: 156
• 912 BUEI Classes: 162
• 92 Bracketing Entropy: 166
• 93 GlivenkoCantelli Preservation: 169
• 94 Donsker Preservation: 172
• 95 Proofs: 173
• 96 Exercises: 176
• 97 Notes: 178
• Bootstrapping Empirical Processes: 179
• 101 The Bootstrap for Donsker Classes: 180
• 1011 An Unconditional Multiplier Central Limit Theorem: 181
• 1012 Conditional Multiplier Central Limit Theorems: 183
• 1013 Bootstrap Central Limit Theorems: 187
• 1014 Continuous Mapping Results: 189
• 102 The Bootstrap for GlivenkoCantelli Classes: 193
• 103 A Simple ZEstimator Master Theorem: 196
• 104 Proofs: 198
• 105 Exercises: 204
• 106 Notes: 205
• Additional Empirical Process Results: 207
• 111 Bounding Moments and Tail Probabilities: 208
• 112 Sequences of Functions: 211
• 113 Contiguous Alternatives: 214
• 114 Sums of Independent but not Identically Distributed Stochastic Processes: 218
• 1142 Bootstrap Results: 222
• 115 Function Classes Changing with n: 224
• 116 Dependent Observations: 227
• 117 Proofs: 230
• 118 Exercises: 233
• The Functional Delta Method: 234
• 122 Examples: 237
• 1222 Integration: 238
• 1223 Product Integration: 242
• 1224 Inversion: 246
• 1225 Other Mappings: 249
• 124 Notes: 250
• ZEstimators: 251
• 131 Consistency: 252
• 132 Weak Convergence: 253
• 1321 The General Setting: 254
• 1323 A Master Theorem and the Bootstrap: 255
• 133 Using the Delta Method: 258
• 134 Exercises: 262
• MEstimators: 263
• 141 The Argmax Theorem: 264
• 142 Consistency: 266
• 143 Rate of Convergence: 267
• 144 Regular Euclidean MEstimators: 270
• 145 NonRegular Examples: 271
• 1452 Monotone Density Estimation: 277
• 146 Exercises: 280
• 147 Notes: 282
• Case Studies II: 283
• 152 The TwoParameter Cox Score Process: 287
• 153 The Proportional Odds Model Under Right Censoring: 291
• 1531 Nonparametric Maximum Likelihood Estimation: 292
• 1532 Existence: 293
• 1533 Consistency: 295
• 1534 Score and Information Operators: 297
• 1535 Weak Convergence and Bootstrap Validity: 301
• 154 Testing for a Changepoint: 303
• 155 Large p Small n Asymptotics for Microarrays: 306
• 1551 Assessing P Value Approximations: 308
• 1552 Consistency of Marginal Empirical Distribution Functions: 309
• 1553 Inference for Marginal Sample Means: 312
• 156 Exercises: 314
• 157 Notes: 315
• Semiparametric Inference: 317
• Introduction to Semiparametric Inference: 319
• Preliminaries for Semiparametric Inference: 322
• 172 Hilbert Spaces: 324
• 173 More on Banach Spaces: 328
• 174 Exercises: 332
• Semiparametric Models and Efficiency: 333
• 182 Efficiency: 337
• 183 Optimality of Tests: 342
• 184 Proofs: 345
• 185 Exercises: 346
• 186 Notes: 347
• Efficient Inference for FiniteDimensional Parameters: 349
• 191 Efficient Score Equations: 350
• 192 Profile Likelihood and LeastFavorable Submodels: 351
• 1921 The Cox Model for Right Censored Data: 352
• 1922 The Proportional Odds Model for Right Censored Data: 353
• 1923 The Cox Model for Current Status Data: 355
• 1924 Partly Linear Logistic Regression: 356
• 193 Inference: 357
• 1932 The Profile Sampler: 363
• 1933 The Penalized Profile Sampler: 369
• 1934 Other Methods: 371
• 194 Proofs: 373
• 195 Exercises: 376
• 196 Notes: 377
• Efficient Inference for InfiniteDimensional Parameters: 378
• 202 Inference: 387
• 2022 The Piggyback Bootstrap: 389
• 2023 Other Methods: 393
• 203 Exercises: 395
• Semiparametric MEstimation: 397
• 211 Semiparametric Mestimators: 399
• 2112 General Scheme for Semiparametric MEstimators: 401
• 2113 Consistency and Rate of Convergence: 402
• 212 Weighted MEstimators and the Weighted Bootstrap: 407
• 213 Entropy Control: 410
• 214 Examples Continued: 412
• 2142 Binary Regression Under Misspecified Link Function Example 2 Continued: 415
• 2143 Mixture Models Example 3 Continued: 418
• 215 Penalized Mestimation: 420
• 2152 Two Other Examples: 422
• 216 Exercises: 423
• Case Studies III: 424
• 221 The Proportional Odds Model Under Right Censoring Revisited: 426
• 222 Efficient Linear Regression: 430
• 223 Temporal Process Regression: 436
• 224 A Partly Linear Model for Repeated Measures: 444
• 225 Proofs: 453
• 226 Exercises: 456
• 227 Notes: 457
• References: 459
• Author Index: 470
• List of Symbols: 473
• Subject Index: 477

Bibliographic information

• Title: Introduction to Empirical Processes and Semiparametric Inference; Springer Series in Statistics
• Author: Michael R. Kosorok
• Edition: illustrated
• Publisher: Springer, 2007
• ISBN: 0387749780, 9780387749785
• Length: 483 pages
• Subjects: Mathematics / Probability & Statistics / General; Mathematics / Probability & Statistics / Stochastic Processes