文件名称：Robust Statistics
文件大小：1.71MB
文件格式：DJVU
更新时间：2021-12-25 15:27:59
鲁棒性 统计学
1 GENERALITIES 1 1.1 Why Robust Procedures? 1 1.2 What Should a Robust Procedure Achieve? 5 13 Qualitative Robustness, 7 1.4 Quantitative Robustness, 10 1.5 Infinitesimal Aspects, 13 1.6 Optimal Robustness, 16 1.7 Computation of Robust Estimates, 17 2 THE WEAK TOPOLOGY AND ITS METRIZATION 20 2.1 General Remarks, 20 2.2 The Weak Topology, 20 23 Levy and Prohorov Metrics, 25 2.4 The Bounded Lipschitz Metric, 29 2.5 Frechet and Gateaux Derivatives, 34 2.6 Hampel's Theorem, 40 3 THE BASIC TYPES OF ESTIMATES 43 3.1 General Remarks, 43 3.2 Maximum Likelihood Type Estimates (M-Estimates), 43 33 Linear Combinations of Order Statistics (L-Estimates), 55 3.4 Estimates Derived from Rank Tests (/?-Estimates), 61 3.5 Asymptotically Efficient M-, L-, and /^-Estimates, 68 4 ASYMPTOTIC MINIMAX THEORY FOR ESTIMATING A LOCATION PARAMETER 73 4.1 General Remarks, 73 4.2 Minimax Bias, 74 43 Minimax Variance: Preliminaries, 76 4.4 Distributions Minimizing Fisher Information, 77 vii Vlll CONTENTS 4.5 Determination of Fo by Variational Methods, 82 4.6 Asymptotically Minimax M-Estimates, 94 4.7 On the Minimax Property for L- and /^-Estimates, 97 4.8 Descending M-Estimates, 100 4.9 Questions of Asymmetric Contamination, 104 5 SCALE ESTIMATES 107 5.1 General Remarks, 107 5.2 M-Estimates of Scale, 109 53 L-Estimates of Scale, 110 5.4 /^-Estimates of Scale, 114 5.5 Asymptotically Efficient Scale Estimates, 116 5.6 Distributions Minimizing Fisher Information for Scale, 118 5.7 Minimax Properties, 122 6 MULTIPARAMETER PROBLEMS, IN PARTICULAR JOINT ESTIMATION OF LOCATION AND SCALE 127 6.1 General Remarks, 127 6.2 Consistency of M-Estimates, 127 63 Asymptotic Normality of M-Estimates, 132 6.4 Simultaneous M-Estimates of Location and Scale, 135 6.5 M-Estimates with Preliminary Estimates of Scale, 140 6.6 Quantitative Robustness Properties of Simultaneous Estimates for Location and Scale, 141 6.7 The Computation of M-Estimates, 146 6.8 Studentizing, 148 7 REGRESSION 153 7.1 General Remarks, 153 7.2 The Classical Linear Least Squares Case, 155 73 Robustizing the Least Squares Approach, 162 7.4 Asymptotics of Robust Regression Estimates, 164 7.5 Conjectures and Empirical Results, 170 7.6 Asymptotic Covariances and Their Estimation, 172 7.7 Concomitant Scale Estimates, 175 7.8 Computation of Regression M-Estimates, 179 7.9 Moderate Leverage Points, 192 7.10 Analysis of Variance, 195 CONTENTS IX 8 ROBUST CO VARIANCE AND CORRELATION MATRICES 199 8.1 General Remarks, 199 8.2 Estimation of Matrix Elements through Robust Variances, 202 83 Estimation of Matrix Elements through Robust Correlation, 204 8.4 An Affinely Invariant Approach, 211 8.5 Estimates Determined by Implicit Equations, 213 8.6 Existence and Uniqueness of Solutions, 215 8.7 Influence Functions and Qualitative Robustness, 223 8.8 Consistency and Asymptotic Normality, 226 8.9 Breakdown Point, 227 8.10 Least Informative Distributions, 229 8.11 Some Notes on Computation, 237 9 ROBUSTNESS OF DESIGN 243 9.1 General Remarks, 243 9.2 Minimax Global Fit, 243 93 Minimax Slope, 251 10 EXACT FINITE SAMPLE RESULTS 253 10.1 General Remarks, 253 10.2 Lower and Upper Probabilities and Capacities, 254 103 Robust Tests, 264 10.4 Sequential Tests, 273 10.5 The Neyman-Pearson Lemma for 2-Alternating Capacities, 275 10.6 Estimates Derived from Tests, 278 10.7 Minimax Interval Estimates, 282 11 MISCELLANEOUS TOPICS 286 11.1 Hampel's Extremal Problem, 286 11.2 Shrinking Neighborhoods, 290 REFERENCES 294 INDEX 301