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
