Robust statistics

371 Pages·2009·6.11 MB·English

by  Peter J. Huber

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ROBUSSTT ATISTICS
SecondE dition
PeterH uJb.e r
ProfesosfoS rt atisrteitcisr,e d
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ElvezMi.oR onchetti
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WILEY
A JOHNW ILE&Y SONSI,N C.P,U BLICATION
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CONTENTS
Preface xiii
Preface to First Edition
XV
1 Generalities 1
1.1 Why Robust Procedures? 1
1.2 What Should a Robust Procedure Achieve? 5
1.2.1 Robust, Nonparametric, and Distribution-Free 6
1.2.2 Adaptive Procedures 7
1.2.3 Resistant Procedures 8
1.2.4 Robustness versus Diagnostics 8
1.2.5 Breakdown point 8
1.3 Qualitative Robustness 9
1.4 Quantitative Robustness 11
1.5 Infinitesimal Aspects 14
1.6 Optimal Robustness 17
1.7 Performance Comparisons 18
vii
CONTENTS
Viii
1.8 Computation of Robust Estimates 18
1.9 Limitations to Robustness Theory 20
2 The Weak Topology and its Metrization 23
2.1 General Remarks 23
2.2 The Weak Topology 23
2.3 Levy and Prohorov Metrics 27
2.4 The Bounded Lipschitz Metric 32
2.5 Frechet and Gateaux Derivatives 36
2.6 Hampel's Theorem 41
3 The Basic Types of Estimates 45
3.1 General Remarks 45
3.2 Maximum Likelihood Type Estimates (M -Estimates) 46
3.2.1 Influence Function of M -Estimates 47
3.2.2 Asymptotic Properties of M-Estimates 48
3.2.3 Quantitative and Qualitative Robustness of M-
Estimates 53
3.3 Linear Combinations of Order Statistics (£-Estimates) 55
3.3.1 Influence Function of £-Estimates 56
3.3.2 Quantitative and Qualitative Robustness of £-Estimates 59
3.4 Estimates Derived from Rank Tests (R-Estimates) 60
3.4.1 Influence Function of R-Estimates 62
3.4.2 Quantitative and Qualitative Robustness of R-Estimates 64
3.5 Asymptotically Efficient M-, L-, and R-Estimates 67
4 Asymptotic Minimax Theory for Estimating Location 71
4.1 General Remarks 71
4.2 Minimax Bias 72
4.3 Minimax Variance: Preliminaries 74
4.4 Distributions Minimizing Fisher Information 76
4.5 Determination of F0 by Variational Methods 81
4.6 Asymptotically Minimax M -Estimates 91
4.7 On the Minimax Property for L-and R-Estimates 95
4.8 Redescending M -Estimates 97
4.9 Questions of Asymmetric Contamination 101
CONTENTS ix
5 Scale Estimates 105
5.1 General Remarks 105
5.2 M-Estimates of Scale 107
5.3 L-Estimates of Scale 109
5.4 R-Estimates of Scale 112
5.5 Asymptotically Efficient Scale Estimates 114
5.6 Distributions Minimizing Fisher Information for Scale 115
5.7 Minimax Properties 119
6 Multiparameter Problems-in Particular Joint Estimation
of Location and Scale 125
6.1 General Remarks 125
6.2 Consistency of M -Estimates 126
6.3 Asymptotic Normality of M-Estimates 130
6.4 Simultaneous M -Estimates of Location and Scale 133
6.5 M -Estimates with Preliminary Estimates of Scale 137
6.6 Quantitative Robustness of Joint Estimates of Location and Scale 139
6.7 The Computation of M-Estimates of Scale 143
6.8 Studentizing 145
7 Regression 149
7.1 General Remarks 149
7.2 The Classical Linear Least Squares Case 154
7.2.1 Residuals and Outliers 158
7.3 Robustizing the Least Squares Approach 160
7.4 Asymptotics of Robust Regression Estimates 163
7.4.1 The Cases hp2 0 and hp 0 164
___. ___.
7.5 Conjectures and Empirical Results 168
7.5.1 Symmetric Error Distributions 168
7.5.2 The Question of Bias 168
7.6 Asymptotic Covariances and Their Estimation 170
7.7 Concomitant Scale Estimates 172
7.8 Computation of Regression M -Estimates 175
7.8.1 The Scale Step 176
7.8.2 The Location Step with Modified Residuals 178
7.8.3 The Location Step with Modified Weights 179