Robust Statistics
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"A comprehensive introduction and discussion on the formal mathematical background behind qualitative and quantitative robustness is provided, and subsequent chapters delve into basic types of scale estimates, asymptotic minimax theory, regression, robust covariance, and robust design . . . it also serves as a valuable reference for researchers and practitioners who wish to study the statistical research associated with robust statistics" (Mathematical Reviews, 2010)More details
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Content
Preface to First Edition.
1. Generalities.
1.1 Why Robust Procedures?
1.2 What Should a Robust Procedure Achieve?
1.3 Qualitative Robustness.
1.4 Quantitative Robustness.
1.5 Infinitesimal Aspects.
1.6 Optimal Robustness.
1.7 Computation of Robust Estimates.
1.8 Limitations to Robustness Theory.
2. The Weak Topology and its Metrization.
2.1 General Remarks.
2.2 The Weak Topology.
2.3 Lévy and Prohorov Metrics.
2.4 The Bounded Lipschitz Metric.
2.5 Fréechet and Gâteaux Derivatives.
2.6 Hampel's Theorem.
3. The Basic Types of Estimates.
3.1 General Remarks.
3.2 Maximum Likelihood Type Estimates (MEstimates).
3.3 Linear Combinations of Order Statistics (LEstimates).
3.4 Estimates Derived from Rank Tests (REstimates).
3.5 Asymptotically Efficient M, L, and REstimates.
4. Asymptotic Minimax Theory for Estimating Location.
4.1 General Remarks.
4.2 Minimax Bias.
4.3 Minimax Variance: Preliminaries.
4.4 Distributions Minimizing Fisher Information.
4.5 Determination of F0 by Variational Methods.
4.6 Asymptotically Minimax MEstimates.
4.7 On the Minimax Property for Land REstimates.
4.8 Redescending MEstimates.
4.9 Questions of Asymmetric Contamination.
5. Scale Estimates.
5.1 General Remarks.
5.2 MEstimates of Scale.
5.3 LEstimates of Scale.
5.4 REstimates of Scale.
5.5 Asymptotically Efficient Scale Estimates.
5.6 Distributions Minimizing Fisher Information for Scale.
5.7 Minimax Properties.
6. Multiparameter Problems, in Particular Joint Estimation of Location and Scale.
6.1 General Remarks.
6.2 Consistency of MEstimates.
6.3 Asymptotic Normality of MEstimates.
6.4 Simultaneous MEstimates of Location and Scale.
6.5 MEstimates with Preliminary Estimates of Scale.
6.6 Quantitative Robustness of Joint Estimates of Location and Scale.
6.7 The Computation of MEstimates of Scale.
6.8 Studentizing.
7. Regression.
7.1 General Remarks.
7.2 The Classical Linear Least Squares Case.
7.2.1 Residuals and Outliers.
7.3 Robustizing the Least Squares Approach.
7.4 Asymptotics of Robust Regression Estimates.
7.5 Conjectures and Empirical Results.
7.6 Asymptotic Covariances and Their Estimation.
7.7 Concomitant Scale Estimates.
7.8 Computation of Regression MEstimates.
7.9 The Fixed Carrier Case: what size hi?
7.10 Analysis of Variance.
7.11 L1estimates and Median Polish.
7.12 Other Approaches to Robust Regression.
8. Robust Covariance and Correlation Matrices.
8.1 General Remarks.
8.2 Estimation of Matrix Elements Through Robust Variances.
8.3 Estimation of Matrix Elements Through Robust Correlation.
8.4 An Affinely Equivariant Approach.
8.5 Estimates Determined by Implicit Equations.
8.6 Existence and Uniqueness of Solutions.
8.7 Influence Functions and Qualitative Robustness.
8.8 Consistency and Asymptotic Normality.
8.9 Breakdown Point.
8.10 Least Informative Distributions.
8.11 Some Notes on Computation.
9. Robustness of Design.
9.1 General Remarks.
9.2 Minimax Global Fit.
9.3 Minimax Slope.
10. Exact Finite Sample Results.
10.1 General Remarks.
10.2 Lower and Upper Probabilities and Capacities.
10.3 Robust Tests.
10.4 Sequential Tests.
10.5 The NeymanPearson Lemma for 2Alternating Capacities.
10.6 Estimates Derived From Tests.
10.7 Minimax Interval Estimates.
11. Finite Sample Breakdown Point.
11.1 General Remarks.
11.2 Definition and Examples.
11.3 Infinitesimal Robustness and Breakdown.
11.4 Malicious versus Stochastic Breakdown.
12. Infinitesimal Robustness.
12.1 General Remarks.
12.2 Hampel's Infinitesimal Approach.
12.3 Shrinking Neighborhoods.
13. Robust Tests.
13.1 General Remarks.
13.2 Local Stability of a Test.
13.3 Tests for General Parametric Models in the Multivariate Case.
13.4 Robust Tests for Regression and Generalized Linear Models.
14. Small Sample Asymptotics.
14.1 General Remarks.
14.2 Saddlepoint Approximation for the Mean.
14.3 Saddlepoint Approximation of the Density of Mestimators.
14.4 Tail Probabilities.
14.5 Marginal Distributions.
14.6 Saddlepoint Test.
14.7 Relationship with Nonparametric Techniques.
15. Bayesian Robustness.
15.1 General Remarks.
15.2 Disparate Data and Problems with the Prior.
15.3 Maximum Likelihood and Bayes Estimates.
15.4 Some Asymptotic Theory.
15.5 Minimax Asymptotic Robustness Aspects.
15.6 Nuisance Parameters.
15.7 Why there is no Finite Sample Bayesian Robustness Theory.
References.
Index.
