ROBUST STATISTICAL METHODS with R

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Jana Jurecková
1 Mathematical tools of robustness 5
1.1 Statistical model 5
1.2 Illustration on statistical estimation 8
1.3 Statistical functional 9
1.4 Fisher consistency 11
1.5 Some distances of probability measures 12
1.6 Relations between distances 13
1.7 Differentiable statistical functionals 14
1.8 Gˆateau derivative 15
1.9 Fr´echet derivative 17
1.10 Hadamard (compact) derivative 18
1.11 Large sample distribution of empirical functional 18
1.12 Computation and software notes 19
1.13 Problems and complements 23
2 Basic characteristics of robustness 27
2.1 Influence function 27
2.2 Discretized form of influence function 28
2.3 Qualitative robustness 30
v
vi CONTENTS
2.4 Quantitative characteristics of robustness based on influence
function 32
2.5 Maximum bias 33
2.6 Breakdown point 35
2.7 Tail–behavior measure of a statistical estimator 36
2.8 Variance of asymptotic normal distribution 41
2.9 Problems and complements 41
3 Robust estimators of real parameter 43
3.1 Introduction 43
3.2 M-estimators 43
3.3 M-estimator of location parameter 45
3.4 Finite sample minimax property of M-estimator 54
3.5 Moment convergence of M-estimators 58
3.6 Studentized M-estimators 61
3.7 L-estimators 63
3.8 Moment convergence of L-estimators 70
3.9 Sequential M- and L-estimators 72
3.10 R-estimators 74
3.11 Numerical illustration 77
3.12 Computation and software notes 80
3.13 Problems and complements 83
4 Robust estimators in linear model 85
4.1 Introduction 85
4.2 Least squares method 87
4.3 M-estimators 94
4.4 GM-estimators 98
4.5 S-estimators and MM-estimators 100
4.6 L-estimators, regression quantiles 101
4.7 Regression rank scores 104
4.8 Robust scale statistics 106
CONTENTS vii
4.9 Estimators with high breakdown points 109
4.10 One-step versions of estimators 110
4.11 Numerical illustrations 112
4.12 Computation and software notes 115
4.13 Problems and complements 126
5 Multivariate location model 129
5.1 Introduction 129
5.2 Multivariate M-estimators of location and scatter 129
5.3 High breakdown estimators of multivariate location and scatter 132
5.4 Admissibility and shrinkage 133
5.5 Numerical illustrations and software notes 134
5.6 Problems and complements 139
6 Some large sample properties of robust procedures 141
6.1 Introduction 141
6.2 M-estimators 142
6.3 L-estimators 144
6.4 R-estimators 146
6.5 Interrelationships of M-, L- and R-estimators 146
6.6 Minimaximally robust estimators 150
6.7 Problems and complements 153
7 Some goodness-of-fit tests 155
7.1 Introduction 155
7.2 Tests of normality of the Shapiro-Wilk type with nuisance
regression and scale parameters 155
7.3 Goodness-of-fit tests for general distribution with nuisance
regression and scale 158
7.4 Numerical illustration 160
7.5 Computation and software notes 166

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关键词：Statistical statistica statistic Statist Methods Methods Statistical robust

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