Robust statistics

Cover......Page 1
Title......Page 4
Copyright Page......Page 5
Contents......Page 8
Preface......Page 14
Preface to the First Edition......Page 16
1.1 Why Robust Procedures......Page 18
1.2 What Should a Robust Procedure Achieve......Page 22
1.2.1  Robust, Nonparametric, and Distribution-Free......Page 23
1.2.2  Adaptive Procedures......Page 24
1.2.5  Breakdown point......Page 25
1.3 Qualitative Robustness......Page 26
1.4 Quantitative Robustness......Page 28
1.5 Infinitesimal Aspects......Page 31
1.6 Optimal Robustness......Page 34
1.8 Compu ta tion of Robust Estimates......Page 35
1.9 Limitations to Robus tness Theory......Page 37
2.2 The Weak Topology......Page 40
2.3 Levy and Prohorov Metrics......Page 44
2.4 The Bounded Lipschitz Metric......Page 49
2.5 Frechet and Ga teaux Derivatives......Page 53
2.6 Hampel's Theorem......Page 58
3.1  General Remarks......Page 62
3.2  Maximum Likelihood Type Es timates (M -Estimates)......Page 63
3.2.1  Influence Function of M-Estimates......Page 64
3.2.2  Asymptotic Properties of M-Estimates......Page 65
3.2.3  Quantitative and Qualitative Robustness of M-Estimates......Page 70
3.3  Linear Combinations of  Order Statistics (L-Estimates)......Page 72
3.3.1  Influence Function of  L-Estimates......Page 73
3.3.2  Quantitative and Qualitative Robustness of L-Estimates......Page 76
3.4  Estimates Derived from Rank Tests (R-Estimates)......Page 77
3.4.1  Influence Function of R-Estimates......Page 79
3.4.2  Quantitative and Qualitative Robustness of R-Estimates......Page 81
3.5  Asymptotically Efficient M-, L-, and R-Estimates......Page 84
4.1 General Remarks......Page 88
4.2 Minimax Bias......Page 89
4.3 Minimax Variance: Preliminaries......Page 91
4.4 Dis tributions Minimizing Fisher Information......Page 93
4.5 Determination of F_0 by Variational Methods......Page 98
4.6 Asymptotically Minimax M -Estimates......Page 108
4.7 On the Minimax Property for L- and R-Estimates......Page 112
4.8 Redescending M-Estimates......Page 114
4.9 Questions of Asymmetric Contamination......Page 118
5.1  General Remarks......Page 122
5.2  M-Estimates of Scale......Page 124
5.3  L-Estimates of Scale......Page 126
5.4  R-Estimates of Scale......Page 129
5.5  Asymptotically Efficient  Scale Estimates......Page 131
5.6  Distributions Minimizing Fisher Information for Scale......Page 132
5.7  Minimax Properties......Page 136
6.1  General Remarks......Page 142
6.2  Consistency of M-Estimates......Page 143
6.3  Asymptotic Normality of M-Estimates......Page 147
6.4  Simultaneous M-Estimates of Location and Scale......Page 150
6.5  M-Estimates with Preliminary Estimates of Scale......Page 154
6.6  Quantitative Robustness of Joint Estimates of Location and Scale......Page 156
6.7  The Computation of M-Estimates of Scale......Page 160
6.8  Studentizing......Page 162
7.1  General Remarks......Page 166
7.2  The Classical Linear Least Squares Case......Page 171
7.2.1  Residuals and Outliers......Page 175
7.3  Robus t izing the Least Squares Approach......Page 177
7.4  Asymptotics of Robust Regression Estimates......Page 180
7.4.1  The Cases hp^2 -> 0 and hp -> 0......Page 181
7.5.2  The Question of Bias......Page 185
7.6  Asymptotic Covariances and Their Estimation......Page 187
7.7  Concomitant Scale Es timates......Page 189
7.8  Computation of Regression M-Estimates......Page 192
7.8.1  The Scale Step......Page 193
7.8.2  The Location Step with Modified Residuals......Page 195
7.8.3  The Location Step with Modified Weights......Page 196
7.9  The Fixed Carrier Case: What Size hi......Page 203
7.10  Analysis of Variance......Page 207
7.11  L1  -estimates and Median Polish......Page 210
7.12  Other Approaches to Robust Regression......Page 212
8.1  General Remarks......Page 216
8.2  Estimation of Matrix Elements Through Robust Variances......Page 220
8.3  Estimation of Matrix Elements Through Robust Correlation......Page 221
8.4  An Affinely Equivariant Approach......Page 227
8.5  Estimates Determined by Implicit Equations......Page 229
8.6.1  The Scatter Estimate V......Page 231
8.6.2  The Location Estimate t......Page 236
8.7  Influence Functions and Qualitative Robustness......Page 237
8.8  Consistency and Asymptotic Normality......Page 240
8.9  Breakdown Point......Page 241
8.10.1  Location......Page 242
8.10.2  Covariance......Page 244
8.11  Some Notes on Computation......Page 250
9.1  General Remarks......Page 256
9.2  Minimax Global Fit......Page 257
9.3  Minimax Slope......Page 263
10.1  General Remarks......Page 266
10.2  Lower and Upper Probabilities and Capacities......Page 267
10.2.1  2-Monotone and 2-Altemating Capacities......Page 272
10.2.2  Monotone and Al ternating Capacities of  Infinite Order......Page 275
10.3  Robust Tests......Page 276
10.3.1  Particular Cases......Page 282
10.4  Sequential Tests......Page 284
10.5  The Neyman-Pearson Lemma for 2-Alternating Capacities......Page 286
10.6  Estimates Derived From Tests......Page 289
10.7  Minimax Interval Estimates......Page 293
11.1  General Remarks......Page 296
11.2  Definition and Examples......Page 298
11.2.2  Multidimensional Estimators of Location......Page 300
11.2.3  Structured Problems: Linear Models......Page 301
11.3  Infinitesimal Robustness and Breakdown......Page 303
11.4  Malicious versus Stochastic Breakdown......Page 304
12.1  General Remarks......Page 306
12.2  Hampel 's Infinitesimal Approach......Page 307
12.3  Shrinking Neighborhoods......Page 311
13.1  General Remarks......Page 314
13.2  Local Stability of a Test......Page 315
13.3  Tests for General Parametric Models in the Multivariate Case......Page 318
13.4  Robust Tests for Regression and Generalized Linear Models......Page 321
14.1  General Remarks......Page 324
14.2  Saddlepoint Approximation for the Mean......Page 325
14.3  Saddlepoint Approximation of  the Density of M-estimators......Page 328
14.4  Tail Probabilities......Page 330
14.5  Marginal Distributions......Page 331
14.6  Saddlepoint Test......Page 333
14.7  Relationship with Nonparametric Techniques......Page 334
14.8  Appendix......Page 338
15.1  General Remarks......Page 340
15.2  Disparate Data and Problems with the Prior......Page 343
15.3  Maximum Likelihood and Bayes Estimates......Page 344
15.4  Some Asymptotic Theory......Page 346
15.5  Minimax Asymptotic Robustness Aspects......Page 347
15.7  Why  there is no Finite Sample Bayesian Robustness Theory......Page 348
References......Page 350
Index......Page 362