Introduction to Empirical Processes and Semiparametric Inference

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Springer, Dec 29, 2007 - Mathematics - 483 pages
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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.
 

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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
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

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
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
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