Robust Statistical Methods With R

Robust Statistical Methods With R......Page 2
Preface......Page 8
Authors......Page 10
Contents......Page 4
Introduction......Page 11
1.1 Statistical model......Page 14
1.2 Illustration on statistical estimation......Page 17
1.3 Statistical functional......Page 18
1.4 Fisher consistency......Page 20
1.5 Some distances of probability measures......Page 21
1.6 Relations between distances......Page 22
1.7 Differentiable statistical functionals......Page 23
1.8 Gateau derivative......Page 24
1.9 Fréchet derivative......Page 26
1.11 Large sample distribution of empirical functional......Page 27
1.12 Computation and software notes......Page 28
1.13 Problems and complements......Page 32
2.1 Influence function......Page 35
2.2 Discretized form of influence function......Page 36
2.3 Qualitative robustness......Page 38
2.4 Quantitative characteristics of robustness based on influence function......Page 40
2.5 Maximum bias......Page 41
2.6 Breakdown point......Page 43
2.7 Tail– behavior measure of a statistical estimator......Page 44
2.9 Problems and complements......Page 49
3.2 M- estimators......Page 51
3.2.1 Influence function of M- estimator......Page 52
3.3 M- estimator of location parameter......Page 53
3.3.1 Breakdown point of M- estimator of location parameter......Page 54
3.3.2 Choice of function v......Page 56
3.3.3 Other choices of  v......Page 57
3.4 Finite sample minimax property of M- estimator......Page 62
3.5 Moment convergence of M- estimators......Page 66
3.6 Studentized M- estimators......Page 69
3.7 L- estimators......Page 71
3.7.1 Influence function of L- estimator......Page 73
3.7.2 Breakdown point of L- estimator......Page 74
3.8 Moment convergence of L- estimators......Page 78
3.9 Sequential M- and L- estimators......Page 80
3.10 R- estimators......Page 82
3.11 Numerical illustration......Page 85
3.12 Computation and software notes......Page 88
3.13 Problems and complements......Page 91
4.1 Introduction......Page 93
4.2 Least squares method......Page 95
4.3 M- estimators......Page 102
4.3.1 Influence function of M- estimator with random matrix......Page 104
4.3.2 Large sample distribution of the M- estimator with nonrandom matrix......Page 105
4.4 GM- estimators......Page 106
4.5 S- estimators and MM- estimators......Page 108
4.6 L- estimators, regression quantiles......Page 109
4.7 Regression rank scores......Page 112
4.8 Robust scale statistics......Page 114
4.9 Estimators with high breakdown points......Page 117
4.10 One- step versions of estimators......Page 118
4.11 Numerical illustrations......Page 120
4.12 Computation and software notes......Page 123
4.13 Problems and complements......Page 134
5.2 Multivariate M- estimators of location and scatter......Page 137
5.3 High breakdown estimators of multivariate location and scatter......Page 140
5.4 Admissibility and shrinkage......Page 141
5.5 Numerical illustrations and software notes......Page 142
5.6 Problems and complements......Page 147
6.1 Introduction......Page 149
6.2.1 M- estimator of general scalar parameter......Page 150
6.2.2 M- estimators of location parameter......Page 151
6.3 L- estimators......Page 152
6.5 Interrelationships of M-, L- and R- estimators......Page 154
6.5.1 M- and L- estimators......Page 155
6.5.2 M- and R- estimators......Page 157
6.6 Minimaximally robust estimators......Page 158
6.6.1 Minimaximally robust M-, L- and R- estimators......Page 159
6.7 Problems and complements......Page 161
7.2 Tests of normality of the Shapiro- Wilk type with nuisance regression and scale parameters......Page 163
7.3 Goodness- of- fit tests for general distribution with nuisance......Page 166
7.4.1 Comparison of tests for testing normality......Page 168
7.4.2 Testing for nonnormal distributions......Page 171
7.5 Computation and software notes......Page 174
APPENDIX A- R system......Page 181
A. 1.1 Data creation, selection, and manipulation......Page 182
A. 1.4 Math......Page 184
A. 1.5 Distributions......Page 185
A. 1.6 Statistics, optimization, and model fitting......Page 186
A. 1.7 Plotting......Page 187
A. 1.8 Programming......Page 188