INTRODUCTION TO EMPIRICAL PROCESSES AND SEMIPARAMETRIC INFERENCE
CONTENTS I OVERVIEW INTRODUCTION AN OVERVIEW OF EMPIRICAL PROCESSES THE MAIN FEATURES EMPIRICAL PROCESS TECHNIQUES STOCHASTIC CONVERGENCE ENTROPY FOR GLIVENKO-CANTELLI AND DONSKER THEOREMS BOOTSTRAPPING EMPIRICAL PROCESSES THE FUNCTIONAL DELTA METHOD Z-ESTIMATORS M-ESTIMATORS OTHER TOPICS EXERCISES NOTES OVERVIEW OF SEMIPARAMETRIC INFERENCE SEMIPARAMETRIC MODELS AND E?CIENCY SCORE FUNCTIONS AND ESTIMATING EQUATIONS MAXIMUM LIKELIHOOD ESTIMATION CASE STUDIES I LINEAR REGRESSION MEAN ZERO RESIDUALS MEDIAN ZERO RESIDUALS COUNTING PROCESS REGRESSION THE GENERAL CASE THE COX MODEL THE KAPLAN-MEIER ESTIMATOR E?CIENT ESTIMATING EQUATIONS FOR REGRESSION SIMPLE LINEAR REGRESSION A POISSON MIXTURE REGRESSION MODEL PARTLY LINEAR LOGISTIC REGRESSION II EMPIRICAL PROCESSES INTRODUCTION TO EMPIRICAL PROCESSES PRELIMINARIES FOR EMPIRICAL PROCESSES METRIC SPACES OUTER EXPECTATION LINEAR OPERATORS AND FUNCTIONAL DI?ERENTIATION PROOFS STOCHASTIC PROCESSES IN METRIC SPACES WEAK CONVERGENCE GENERAL THEORY SPACES OF BOUNDED FUNCTIONS OTHER MODES OF CONVERGENCE EMPIRICAL PROCESS METHODS MAXIMAL INEQUALITIES ORLICZ NORMS AND MAXIMA MAXIMAL INEQUALITIES FOR PROCESSES THE SYMMETRIZATION INEQUALITY AND MEASURABILITY GLIVENKO-CANTELLI RESULTS DONSKER RESULTS ENTROPY CALCULATIONS UNIFORM ENTROPY VC-CLASSES BUEI CLASSES BRACKETING ENTROPY GLIVENKO-CANTELLI PRESERVATION DONSKER PRESERVATION 10 BOOTSTRAPPING EMPIRICAL PROCESSES 101 THE BOOTSTRAP FOR DONSKER CLASSES 1011 AN UNCONDITIONAL MULTIPLIER CENTRAL LIMIT THEOREM 1012 CONDITIONAL MULTIPLIER CENTRAL LIMIT THEOREMS 1013 BOOTSTRAP CENTRAL LIMIT THEOREMS 1014 CONTINUOUS MAPPING RESULTS 102 THE BOOTSTRAP FOR GLIVENKO-CANTELLI CLASSES 103 A SIMPLE Z-ESTIMATOR MASTER THEOREM 104 PROOFS 105 EXERCISES 106 NOTES 11 ADDITIONAL EMPIRICAL PROCESS RESULTS 111 BOUNDING MOMENTS AND TAIL PROBABILITIES 112 SEQUENCES OF FUNCTIONS 113 CONTIGUOUS ALTERNATIVES 114 SUMS OF INDEPENDENT BUT NOT IDENTICALLY DISTRIBUTED STOCHAS- TIC PROCESSES 1141 CENTRAL LIMIT THEOREMS 1142 BOOTSTRAP RESULTS 115 FUNCTION CLASSES CHANGING WITH N 116 DEPENDENT OBSERVATIONS 117 PROOFS 118 EXERCISES 119 NOTES 12 THE FUNCTIONAL DELTA METHOD 121 MAIN RESULTS AND PROOFS 122 EXAMPLES 1221 COMPOSITION 1222 INTEGRATION 1223 PRODUCT INTEGRATION 1224 INVERSION 1225 OTHER MAPPINGS 123 EXERCISES 124 NOTES 13 Z-ESTIMATORS 131 CONSISTENCY 132 WEAK CONVERGENCE 1321 THE GENERAL SETTING 1322 USING DONSKER CLASSES 1323 A MASTER THEOREM AND THE BOOTSTRAP 133 USING THE DELTA METHOD 134 EXERCISES 135 NOTES 14 M-ESTIMATORS 141 THE ARGMAX THEOREM 142 CONSISTENCY 143 RATE OF CONVERGENCE 144 REGULAR EUCLIDEAN M-ESTIMATORS 145 NON-REGULAR EXAMPLES 1451 A CHANGE-POINT MODEL 1452 MONOTONE DENSITY ESTIMATION 146 EXERCISES 147 NOTES 15 CASE STUDIES II 151 PARTLY LINEAR LOGISTIC REGRESSION REVISITED 152 THE TWO-PARAMETER COX SCORE PROCESS 153 THE PROPORTIONAL ODDS MODEL UNDER RIGHT CENSORING 1531 NONPARAMETRIC MAXIMUM LIKELIHOOD ESTIMATION 1532 EXISTENCE 1533 CONSISTENCY 1534 SCORE AND INFORMATION OPERATORS 1535 WEAK CONVERGENCE AND BOOTSTRAP VALIDITY 154 TESTING FOR A CHANGE-POINT 155 LARGE P SMALL N ASYMPTOTICS FOR MICROARRAYS 1551 ASSESSING P-VALUE APPROXIMATIONS 1552 CONSISTENCY OF MARGINAL EMPIRICAL DISTRIBUTION FUNCTIONS 1553 INFERENCE FOR MARGINAL SAMPLE MEANS 156 EXERCISES 157 NOTES III SEMIPARAMETRIC INFERENCE 16 INTRODUCTION TO SEMIPARAMETRIC INFERENCE 17 PRELIMINARIES FOR SEMIPARAMETRIC INFERENCE 171 PROJECTIONS 172 HILBERT SPACES 173 MORE ON BANACH SPACES 174 EXERCISES 175 NOTES 18 SEMIPARAMETRIC MODELS AND E?CIENCY 181 TANGENT SETS AND REGULARITY 182 E?CIENCY 183 OPTIMALITY OF TESTS 184 PROOFS 185 EXERCISES 186 NOTES 19 E?CIENT INFERENCE FOR FINITE-DIMENSIONAL PARAMETERS 191 E?CIENT SCORE EQUATIONS 192 PRO?LE LIKELIHOOD AND LEAST-FAVORABLE SUBMODELS 1921 THE COX MODEL FOR RIGHT CENSORED DATA 1922 THE PROPORTIONAL ODDS MODEL FOR RIGHT CENSORED DATA 1923 THE COX MODEL FOR CURRENT STATUS DATA 1924 PARTLY LINEAR LOGISTIC REGRESSION 193 INFERENCE 1931 QUADRATIC EXPANSION OF THE PRO?LE LIKELIHOOD 1932 THE PRO?LE SAMPLER 1933 THE PENALIZED PRO?LE SAMPLER 1934 OTHER METHODS 194 PROOFS 195 EXERCISES 196 NOTES 20 E?CIENT INFERENCE FOR IN?NITE-DIMENSIONAL PARAMETERS 201 SEMIPARAMETRIC MAXIMUM LIKELIHOOD ESTIMATION 202 INFERENCE 2021 WEIGHTED AND NONPARAMETRIC BOOTSTRAPS 2022 THE PIGGYBACK BOOTSTRAP 2023 OTHER METHODS 203 EXERCISES 204 NOTES 21 SEMIPARAMETRIC M-ESTIMATION 211 SEMIPARAMETRIC M-ESTIMATORS 2111 MOTIVATING EXAMPLES 2112 GENERAL SCHEME FOR SEMIPARAMETRIC M-ESTIMATORS 2113 CONSISTENCY AND RATE OF CONVERGENCE 2114 ?N CONSISTENCY AND ASYMPTOTIC NORMALITY 212 WEIGHTED M-ESTIMATORS AND THE WEIGHTED BOOTSTRAP 213 ENTROPY CONTROL 214 EXAMPLES CONTINUED 2141 COX MODEL WITH CURRENT STATUS DATA (EXAMPLE 1, CONTINUED) 2142 BINARY REGRESSION UNDER MISSPECI?ED LINK FUNCTION (EXAMPLE 2, CONTINUED) 2143 MIXTURE MODELS (EXAMPLE 3, CONTINUED) 215 PENALIZED M-ESTIMATION 2151 BINARY REGRESSION UNDER MISSPECI?ED LINK FUNCTION (EXAMPLE 2, CONTINUED) 2152 TWO OTHER EXAMPLES 216 EXERCISES 217 NOTES 22 CASE STUDIES III 221 THE PROPORTIONAL ODDS MODEL UNDER RIGHT CENSORING REVISITED 222 E?CIENT LINEAR REGRESSION 223 TEMPORAL PROCESS REGRESSION 224 A PARTLY LINEAR MODEL FOR REPEATED MEASURES 225 PROOFS 226 EXERCISES 227 NOTES REFERENCES