半参数理论与损失数据处理

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1 Introduction to Semiparametric Models . . .  . . . 1
1.1 What Is an Infinite-Dimensional Space? . . . . . . 2
1.2 Examples of Semiparametric Models . .   . . 3
1.3 Semiparametric Estimators . . .   . . . 8
2 Hilbert Space for Random Vectors . .  . . . 11
2.1 The Space of Mean-Zero q-dimensional Random Functions . .  . . . . . . 11
The Dimension of the Space of Mean-Zero Random Functions . .  . . . . . . 12
2.2 Hilbert Space . . . . . . . . . .  . . . . . . 13
2.3 Linear Subspace of a Hilbert Space and the Projection Theorem . .   . . . . . . . 14
3 The Geometry of Influence Functions. . . . . . . . . . . . . . . . . . . . . . 21
3.1 Super-Efficiency . . . . . . .  . . . . . . . 24
3.2 m-Estimators (Quick Review) . . . .  . . . . . . . . . . 29
Estimating the Asymptotic Variance of an m-Estimator . . . . . . 31
3.3 Geometry of Influence Functions for Parametric Models . . . . . . 38
Constructing Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 Efficient Influence Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Asymptotic Variance when Dimension Is Greater than One . . . 43
Geometry of Influence Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Deriving the Efficient Influence Function . . . . . . . . . . . . . . . . . . . 46
3.5 Review of Notation for Parametric Models . . . . . . . . . . . . . . . . . . 49
4 Semiparametric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.1 GEE Estimators for the Restricted Moment Model . . . . . . . . . . 54
Asymptotic Properties for GEE Estimators . . . . . . . . . . . . . . . . . 55
4.2 Parametric Submodels . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . 59
4.3 Influence Functions for Semiparametric RAL Estimators . . . . . . . . . . . . .  . . . . . . . . 61
4.4 Semiparametric Nuisance Tangent Space . . . . . . . . . . . . . . . . . . . 63
Tangent Space for Nonparametric Models. . . . . . . . . . . . . . . . . . . 68
Partitioning the Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5 Semiparametric Restricted Moment Model . . . . . . . . . . . . . . . . . . 73
Influence Functions and the Efficient Influence Function for the Restricted Moment Model. .  . . . . . . . . . 83
The Efficient Influence Function . . . . . . . . . . . . . . . . . 85
A Different Representation for the Restricted Moment Model . . . . . . . . . .   . . 87
Existence of a Parametric Submodel for the Arbitrary Restricted Moment Model . .   . . . 91
4.6 Adaptive Semiparametric Estimators for the Restricted Moment Model . . . . .   . . . . . . 93
Extensions of the Restricted Moment Model . . . . . . . . . . . . . . . . 97
5 Other Examples of Semiparametric Models . . . . . . . . . . . . . . . . 101
5.1 Location-Shift Regression Model . . . . . . . . . . . . . . . . . . . . . . . . . . 101
The Nuisance Tangent Space and Its Orthogonal Complement for the Location-Shift Regression Model .  . . . 103
Semiparametric Estimators for β . . .   . . . . . . . . . . 106
Efficient Score for the Location-Shift Regression Model . . . . . . . 107
Locally Efficient Adaptive Estimators . . . . . .  . . . . . . . . 108
5.2 Proportional Hazards Regression Model with Censored Data . . . . . . . . . . .  . . . . . . 113
The Nuisance Tangent Space. . . . . . . .  . . . . . . . . . . . 117
The Space Λ2s Associated with λC|X(v|x) . . . . . . . . . . . . . . . . . . 117
The Space Λ1s Associated with λ(v) . . . . . . . . . . . . . . . . . . . . . . . 119
Finding the Orthogonal Complement of the Nuisance Tangent Space . . . . . . . . . . . . 120
Finding RAL Estimators for β . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Efficient Estimator . . . . . . . . . . . . . . . .  . . . . . . . . . . 125
5.3 Estimating the Mean in a Nonparametric Model . . . . . . . . . . . . . 125
5.4 Estimating Treatment Difference in a Randomized
Pretest-Posttest Study or with Covariate Adjustment . . . . . . . . 126
The Tangent Space and Its Orthogonal Complement . . . . . . . . . 129
5.5 Remarks about Auxiliary Variables . . . . . . . . . . . . . . . . . . . . . . . . 133
6 Models and Methods for Missing Data. . . . . . . . . . . . . . . . . . . . . 137
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.2 Likelihood Methods . . . . . . . . . . . . . . 143
6.3 Imputation . . . . . . . . . . . . . . . 144
6.4 Inverse Probability Weighted Complete-Case Estimator . . . . . . 146
6.5 Double Robust Estimator . . . . . . .   . . . . . 147
7 Missing and Coarsening at Random for Semiparametric Models . . .. 151
7.1 Missing and Coarsened Data . . .  . . . . . . 151
Missing Data as a Special Case of Coarsening . . . . . . . . . . . . . . . 153
Coarsened-Data Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.2 The Density and Likelihood of Coarsened Data . . . . . . . . . . . . . . 156
Discrete Data . . . . . . . .   . . . . . 156
Continuous Data . . . . . .  . . . 157
Likelihood when Data Are Coarsened at Random . . . . . . . . . . . . 158
Brief Remark on Likelihood Methods . . . . . . . . . . . . . . . . . . . . . . 160
7.3 The Geometry of Semiparametric Coarsened-Data Models . . . . . . . . 163
The Nuisance Tangent Space Associated with the Full-Data
Nuisance Parameter and Its Orthogonal Complement . . 166
The Logistic Regression Model . . . . .  . . . . . . 179
7.5 Recap and Review of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8 The Nuisance Tangent Space and Its Orthogonal Complement . . .  . . . . . 185
8.1 Models for Coarsening and Missingness . . . . . . . . . . . . . . . . . . . . . 185
Two Levels of Missingness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Monotone and Nonmonotone Coarsening for more than Two Levels . . . . . . . . 186
8.2 Estimating the Parameters in the Coarsening Model . . . . . . . . . 188
MLE for ψ with Two Levels of Missingness . . . . . . . . . . . . . . . . . 188
MLE for ψ with Monotone Coarsening . . . . . . . . . . . . . . . . . . . . . 189
8.3 The Nuisance Tangent Space when Coarsening Probabilities Are Modeled . . . . . . . 190
8.4 The Space Orthogonal to the Nuisance Tangent Space . .  . 192
8.5 Observed-Data Influence Functions . . . . . . . . . . . . . . . . . . . . . . . . 193
8.6 Recap and Review of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
9 Augmented Inverse Probability Weighted Complete-Case Estimators . . . . 199
9.1 Deriving Semiparametric Estimators for β . . . . . . . . . . . . . . . . . . 199
Interesting Fact . . . . . . . . .  . . . . . . . . 206
Estimating the Asymptotic Variance . . . .  . . . . . . 206
9.2 Additional Results Regarding Monotone Coarsening . . . . . . . . . 207
The Augmentation Space Λ2 with Monotone Coarsening. . . . . . 207
9.3 Censoring and Its Relationship to Monotone Coarsening . . . . . . . . . 213
Probability of a Complete Case with Censored Data . . . . . . . . . 216
The Augmentation Space, Λ2, with Censored Data. . . . . . . . . . . 216
Deriving Estimators with Censored Data . . . . . . . . . . . . . . . . . . . 217
9.4 Recap and Review of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
10 Improving Efficiency and Double Robustness with Coarsened Data . .  . 221
10.1 Optimal Observed-Data Influence Function Associated with Full-Data Influence Function . .221
10.2 Improving Efficiency with Two Levels of Missingness . .225
Finding the Projection onto the Augmentation Space . . . . . . . . 226
Adaptive Estimation . . . . . . . . . . . 227
Algorithm for Finding Improved Estimators with Two Levels of Missingness  . . 229
Remarks Regarding Adaptive Estimators . . . . . . . . . . . . . . . . . . . 230
Estimating the Asymptotic Variance . . . . . . . . . . . . . . . . . . . . . . . 233
Double Robustness with Two Levels of Missingness . . . . . . . . . . 234
Remarks Regarding Double-Robust Estimators . . . . . . . . . . . . . . 236
Logistic Regression Example Revisited . . . . . . . . . . . . . . . . . . . . . 236
10.3 Improving Efficiency with Monotone Coarsening . . . . . . . . . . . . . 239
Finding the Projection onto the Augmentation Space . . . . . . . . 239
Adaptive Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
Double Robustness with Monotone Coarsening . . . . . . . . . . . . . . 248
10.4 Remarks Regarding Right Censoring . . . . . . . . . . . . . . . . . . . . . . . 254
10.5 Improving Efficiency when Coarsening Is Nonmonotone . . . .  . 255
Finding the Projection onto the Augmentation Space . . . . . . . . 256
Uniqueness of M−1(·) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
Obtaining Improved Estimators with Nonmonotone Coarsening . .   . . 261
Double Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
10.6 Recap and Review of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
11 Locally Efficient Estimators for Coarsened-Data Semiparametric Models . . . . . 273
11.1 The Observed-Data Efficient Score . . . . . . . . . . . . . . . . . . . . . . . . . 277
Representation 1 (Likelihood-Based) . . . . . . . . . . . . . . . . . . . . . . . 277
Representation 2 (AIPWCC-Based) . . . . . . . . . . . . . . . . . . . . . . . . 278
Relationship between the Two Representations . . . . . . . . . . . . . . 278
M−1 for Monotone Coarsening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
M−1 with Right Censored Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
11.2 Strategy for Obtaining Improved Estimators . . . . . . . . . . . . . . . . 285
11.3 Concluding Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
11.4 Recap and Review of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
12 Approximate Methods for Gaining Efficiency . . . . . . . . . . . . . . 295
12.1 Restricted Class of AIPWCC Estimators . . . . . . . . . . . . . . . . . . . 295
12.2 Optimal Restricted (Class 1) Estimators . . . . . . . . . . . . . . . . . . . . 300
Deriving the Optimal Restricted (Class 1) AIPWCC
Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Estimating the Asymptotic Variance . . . . . . . . . . . . . . . . . . . . . . . 307
12.3 Example of an Optimal Restricted (Class 1) Estimator .   . 309
Modeling the Missingness Probabilities . . . . . . . . . . . . . . . . . . . . . 312
12.4 Optimal Restricted (Class 2) Estimators . . . . . . . . . . . . . . . . . . . . 313
Logistic Regression Example Revisited . . . . . . . . . . . . . . . . . . . . . 319
12.5 Recap and Review of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
13 Double-Robust Estimator of the Average Causal Treatment Effect . . . . . . . . . . . . 323
13.1 Point Exposure Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
13.2 Randomization and Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
13.3 Observational Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
13.4 Estimating the Average Causal Treatment Effect . . . . . . . . . . . . 328
Regression Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
13.5 Coarsened-Data Semiparametric Estimators . . . . . . . . . . . . . . . . 329
Observed-Data Influence Functions . . . . . . . . . . . . . . . . . . . . . . . . 331
Double Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
14 Multiple Imputation: A Frequentist Perspective . . . . . . . . . . . 339
14.1 Full- Versus Observed-Data Information Matrix . . . . . . . . . . . . . 342
14.2 Multiple Imputation . . . . . . . . . . . 344
14.3 Asymptotic Properties of the Multiple-Imputation Estimator  . 346
Stochastic Equicontinuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
14.4 Asymptotic Distribution of the Multiple-Imputation Estimator .  . . 354
14.5 Estimating the Asymptotic Variance . . . . . . . . . . . . . . . . . . . . . . . 362
Consistent Estimator for the Asymptotic Variance . . . . . . . . . . . 365
14.6 Proper Imputation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
Asymptotic Distribution of n1/2( ˆ β∗n− β0) . . . . . . . . . . . . . . . . . . . 367
Rubin’s Estimator for the Asymptotic Variance . . . . . . . . . . . . . 370
Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
14.7 Surrogate Marker Problem Revisited . . . . . . . . . . . . . . . . . . . . . . . 371

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