This book summarizes current knowledge regarding the theory of estimation for semiparametric models with missing data, in an organized and comprehensive manner. It starts with the study of semiparametric methods when there are no missing data. The description of the theory of estimation for semiparametric models is both rigorous and intuitive, relying on geometric ideas to reinforce the intuition and understanding of the theory. These methods are then applied to problems with missing, censored, and coarsened data with the goal of deriving estimators that are as robust and efficient as possible.
Author(s): Anastasios A. Tsiatis
Edition: 1
Year: 2006
Language: English
Pages: 399
Tags: Математика;Теория вероятностей и математическая статистика;Математическая статистика;
Contents......Page 10
Preface......Page 7
1 Introduction to Semiparametric Models......Page 16
1.1 What Is an Infinite-Dimensional Space?......Page 17
Example 1: Restricted Moment Models......Page 18
Example 2: Proportional Hazards Model......Page 22
1.3 Semiparametric Estimators......Page 23
2.1 The Space of Mean-Zero q-dimensional Random Functions......Page 25
The Dimension of the Space of Mean-Zero Random Functions......Page 26
2.2 Hilbert Space......Page 27
Projection Theorem for Hilbert Spaces......Page 28
Example 1: One-Dimensional Random Functions......Page 29
Example 2: q-dimensional Random Functions......Page 30
2.5 Exercises for Chapter 2......Page 33
3 The Geometry of Influence Functions......Page 34
Example Due to Hodges......Page 37
3.2 m-Estimators (Quick Review)......Page 42
Estimating the Asymptotic Variance of an m-Estimator......Page 44
Proof of Theorem 3.2......Page 47
Constructing Estimators......Page 51
3.4 Efficient Influence Function......Page 55
Asymptotic Variance when Dimension Is Greater than One......Page 56
Geometry of Influence Functions......Page 58
Deriving the Efficient Influence Function......Page 59
3.5 Review of Notation for Parametric Models......Page 62
3.6 Exercises for Chapter 3......Page 63
4 Semiparametric Models......Page 65
4.1 GEE Estimators for the Restricted Moment Model......Page 66
Asymptotic Properties for GEE Estimators......Page 67
Example: Log-linear Model......Page 69
4.2 Parametric Submodels......Page 71
4.3 Influence Functions for Semiparametric RAL Estimators......Page 73
4.4 Semiparametric Nuisance Tangent Space......Page 75
Tangent Space for Nonparametric Models......Page 80
Partitioning the Hilbert Space......Page 81
4.5 Semiparametric Restricted Moment Model......Page 85
The Space hebrew omitted[sub(2s)]......Page 89
The Space hebrew omitted[sub(1s)]......Page 91
Influence Functions and the Efficient Influence Function for the Restricted Moment Model......Page 95
The Efficient Influence Function......Page 97
A Different Representation for the Restricted Moment Model......Page 99
Existence of a Parametric Submodel for the Arbitrary Restricted Moment Model......Page 103
4.6 Adaptive Semiparametric Estimators for the Restricted Moment Model......Page 105
Extensions of the Restricted Moment Model......Page 109
4.7 Exercises for Chapter 4......Page 110
5.1 Location-Shift Regression Model......Page 112
The Nuisance Tangent Space and Its Orthogonal Complement for the Location-Shift Regression Model......Page 114
Semiparametric Estimators for β......Page 117
Efficient Score for the Location-Shift Regression Model......Page 118
Locally Efficient Adaptive Estimators......Page 119
5.2 Proportional Hazards Regression Model with Censored Data......Page 124
The Space hebrew omitted[sub(2s)] Associated with λ[(sub(C|X)](v|x)......Page 128
The Space hebrew omitted[sub(1s)] Associated with λ(v)......Page 130
Finding the Orthogonal Complement of the Nuisance Tangent Space......Page 131
Finding RAL Estimators for β......Page 134
5.3 Estimating the Mean in a Nonparametric Model......Page 136
5.4 Estimating Treatment Difference in a Randomized Pretest-Posttest Study or with Covariate Adjustment......Page 137
The Tangent Space and Its Orthogonal Complement......Page 140
5.5 Remarks about Auxiliary Variables......Page 144
5.6 Exercises for Chapter 5......Page 146
6.1 Introduction......Page 148
6.2 Likelihood Methods......Page 154
6.3 Imputation......Page 155
Remarks......Page 156
6.4 Inverse Probability Weighted Complete-Case Estimator......Page 157
6.5 Double Robust Estimator......Page 158
6.6 Exercises for Chapter 6......Page 161
7.1 Missing and Coarsened Data......Page 162
Missing Data as a Special Case of Coarsening......Page 164
Coarsened-Data Mechanisms......Page 165
Discrete Data......Page 167
Continuous Data......Page 168
Likelihood when Data Are Coarsened at Random......Page 169
Brief Remark on Likelihood Methods......Page 171
Examples of Coarsened-Data Likelihoods......Page 172
7.3 The Geometry of Semiparametric Coarsened-Data Models......Page 174
The Nuisance Tangent Space Associated with the Full-Data Nuisance Parameter and Its Orthogonal Complement......Page 177
7.4 Example: Restricted Moment Model with Missing Data by Design......Page 185
The Logistic Regression Model......Page 190
7.5 Recap and Review of Notation......Page 192
7.6 Exercises for Chapter 7......Page 194
Two Levels of Missingness......Page 196
Monotone and Nonmonotone Coarsening for more than Two Levels......Page 197
MLE for ψ with Two Levels of Missingness......Page 199
MLE for ψ with Monotone Coarsening......Page 200
8.3 The Nuisance Tangent Space when Coarsening Probabilities Are Modeled......Page 201
8.4 The Space Orthogonal to the Nuisance Tangent Space......Page 203
8.5 Observed-Data Influence Functions......Page 204
8.6 Recap and Review of Notation......Page 206
8.7 Exercises for Chapter 8......Page 207
9.1 Deriving Semiparametric Estimators for β......Page 209
Estimating the Asymptotic Variance......Page 216
The Augmentation Space hebrew omitted[sub(2)] with Monotone Coarsening......Page 217
9.3 Censoring and Its Relationship to Monotone Coarsening......Page 223
The Augmentation Space, hebrew omitted[sub(2)], with Censored Data......Page 226
Deriving Estimators with Censored Data......Page 227
9.4 Recap and Review of Notation......Page 228
9.5 Exercises for Chapter 9......Page 230
10.1 Optimal Observed-Data Influence Function Associated with Full-Data Influence Function......Page 231
10.2 Improving Efficiency with Two Levels of Missingness......Page 235
Finding the Projection onto the Augmentation Space......Page 236
Adaptive Estimation......Page 237
Algorithm for Finding Improved Estimators with Two Levels of Missingness......Page 239
Remarks Regarding Adaptive Estimators......Page 240
Estimating the Asymptotic Variance......Page 243
Double Robustness with Two Levels of Missingness......Page 244
Logistic Regression Example Revisited......Page 246
Finding the Projection onto the Augmentation Space......Page 249
Adaptive Estimation......Page 253
Double Robustness with Monotone Coarsening......Page 258
Example with Longitudinal Data......Page 261
10.4 Remarks Regarding Right Censoring......Page 264
10.5 Improving Efficiency when Coarsening Is Nonmonotone......Page 265
Finding the Projection onto the Augmentation Space......Page 266
Uniqueness of M[sup(-1)](·)......Page 268
Obtaining Improved Estimators with Nonmonotone Coarsening......Page 271
Double Robustness......Page 275
10.6 Recap and Review of Notation......Page 277
10.7 Exercises for Chapter 10......Page 280
11 Locally Efficient Estimators for Coarsened-Data Semiparametric Models......Page 283
Example: Estimating the Mean with Missing Data......Page 285
Representation 1 (Likelihood-Based)......Page 287
Relationship between the Two Representations......Page 288
M[sup(-1)] for Monotone Coarsening......Page 292
M[sup(-1)] with Right Censored Data......Page 294
11.2 Strategy for Obtaining Improved Estimators......Page 295
Example: Restricted Moment Model with Monotone Coarsening......Page 296
Some Brief Remarks Regarding Robustness......Page 300
11.3 Concluding Thoughts......Page 301
11.4 Recap and Review of Notation......Page 302
11.5 Exercises for Chapter 11......Page 303
12.1 Restricted Class of AIPWCC Estimators......Page 304
12.2 Optimal Restricted (Class 1) Estimators......Page 309
Deriving the Optimal Restricted (Class 1) AIPWCC Estimator......Page 314
Estimating the Asymptotic Variance......Page 316
12.3 Example of an Optimal Restricted (Class 1) Estimator......Page 318
Modeling the Missingness Probabilities......Page 321
12.4 Optimal Restricted (Class 2) Estimators......Page 322
Logistic Regression Example Revisited......Page 328
12.5 Recap and Review of Notation......Page 330
12.6 Exercises for Chapter 12......Page 331
13.1 Point Exposure Studies......Page 332
13.2 Randomization and Causality......Page 335
13.3 Observational Studies......Page 336
Regression Modeling......Page 337
13.5 Coarsened-Data Semiparametric Estimators......Page 338
Observed-Data Influence Functions......Page 340
Double Robustness......Page 345
13.6 Exercises for Chapter 13......Page 346
14 Multiple Imputation: A Frequentist Perspective......Page 347
14.1 Full- Versus Observed-Data Information Matrix......Page 350
14.2 Multiple Imputation......Page 352
14.3 Asymptotic Properties of the Multiple-Imputation Estimator......Page 354
Stochastic Equicontinuity......Page 360
14.4 Asymptotic Distribution of the Multiple-Imputation Estimator......Page 362
14.5 Estimating the Asymptotic Variance......Page 370
Consistent Estimator for the Asymptotic Variance......Page 373
14.6 Proper Imputation......Page 374
Asymptotic Distribution of n[sup(½)](β[sup(*)][sub(n)] – β[sub(0)])......Page 375
Rubin's Estimator for the Asymptotic Variance......Page 378
14.7 Surrogate Marker Problem Revisited......Page 379
How Do We Sample?......Page 381
References......Page 383
D......Page 389
N......Page 390
T......Page 391
