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Robust asymptotic statistics: volume i, Softcover reprint of the original 1st ed. 1994 Volume I Springer Series in Statistics Series

Langue : Anglais

Auteur :

Couverture de l’ouvrage Robust asymptotic statistics: volume i
This volume gives a rigorous account of the asymptotic theory of robust statistics, from the viewpoint of optimally robust functionals and their unbiased estimators and tests. By linking up the local asymptotic minimax criterion with further optimality, the author creates a powerful and unifying notion of robustness. Consequently, this volume gives a self-contained treatment of the main optimality results of asymptotic statistics, and supplies the necessary tools of convex optimization and weak convergence. The volume is an ideal reference for researchers who work in statistics and graduate students with a basic knowledge of probability and statistics. Some of the main topics covered are: von Mises functionals, log-likelihoods, asymptotic statistics, nonparametric statistics, optimal influence curves, the constructional problem, and robust regression.
1: Von Mises Functionals.- 1.1 General Remarks.- 1.2 Regular Differentiations.- 1.3 The Delta Method.- 1.4 M Estimates.- 1.5 Quantiles.- 1.6 L Estimates.- 2: Log Likelihoods.- 2.1 General Remarks.- 2.2 Contiguity and Asymptotic Normality.- 2.3 Differentiable Families.- 2.3.1 Differentiable Arrays.- 2.3.2 Smooth Parametric Families.- 2.3.3 Other Differentiability Notions.- 2.4 Linear Regression.- 3: Asymptotic Statistics.- 3.1 General Remarks.- 3.2 Convolution Representation.- 3.3 Minimax Estimation.- 3.3.1 Normal Mean.- 3.3.2 Asymptotic Minimax Bound.- 3.4 Testing.- 3.4.1 Simple Hypotheses.- 3.4.2 Passage to the Normal Limit.- 3.4.3 One- and Two-Sided Hypotheses.- 3.4.4 Multisided Hypotheses.- 4: Nonparametric Statistics.- 4.1 Introduction.- 4.2 The Nonparametric Setup.- 4.2.1 Full Neighborhood Systems.- 4.2.2 Asymptotically Linear Functionals.- 4.2.3 Asymptotically Linear Estimators.- 4.3 Statistics of Functionals.- 4.3.1 Unbiased Estimation.- 4.3.2 Unbiased Testing.- 4.3.3 Remarks and Criticisms.- 4.4 Restricted Tangent Space.- 5: Optimal Influence Curves.- 5.1 Introduction.- 5.2 Minimax Risk.- 5.3 Oscillation.- 5.3.1 Oscillation/Bias Terms.- 5.3.2 Minimax Oscillation.- 5.4 Robust Asymptotic Tests.- 5.5 Minimax Risk and Oscillation.- 5.5.1 Minimum Trace Subject to Bias Bound.- 5.5.2 Mean Square Error.- 5.5.3 Nonexistence of Strong Solution.- 5.5.4 Equivariance Under Reparametrizations.- 6: Stable Constructions.- 6.1 The Construction Problem.- 6.2 M Equations.- 6.2.1 Location Parameter.- 6.2.2 General Parameter.- 6.3 Minimum Distance.- 6.3.1 MD Functionals.- 6.3.2 MD Estimates.- 6.4 One-Steps.- 6.4.1 Functionals.- 6.4.2 Estimators.- 7: Robust Regression.- 7.1 The Ideal Model.- 7.2 Regression Neighborhoods.- 7.2.1 Errors-in-Variables.- 7.2.2 Error-Free-Variables.- 7.2.3 Translation Invariance.- 7.2.4 Neighborhood Submodels.- 7.2.5 Tangent Subspaces.- 7.3 Conditional Bias.- 7.3.1 General Properties.- 7.3.2 Explicit Terms.- 7.4 Optimal Influence Curves.- 7.4.1 Optimization Problems.- 7.4.2 Auxiliary Results.- 7.4.3 Contamination Optimality.- 7.4.4 Total Variation Optimality.- 7.4.5 Hellinger Optimality.- 7.5 Least Favorable Contamination Curves.- 7.5.1 Hellinger Saddle Points.- 7.5.2 Contamination Saddle Points.- 7.6 Equivariance Under Basis Change.- 7.6.1 Unstandardized Solutions.- 7.6.2 M Standardized Equivariant Solutions.- 7.6.3 Robust Prediction.- Appendix A: Weak Convergence of Measures.- A.1 Basic Notions.- A.2 Convergence of Integrals.- A.3 Smooth Empirical Process.- A.4 Square Integrable Empirical Process.- Appendix B: Some Functional Analysis.- B.1 A Few Facts.- B.2 Lagrange Multipliers.- B.2.1 Neyman—Pearson Lemma.- Appendix C: Complements.- C.1 Parametric Finite-Sample Results.- C.2 Some Technical Results.- C.2.1 Calculus.- C.2.2 Topology.- C.2.3 Matrices.

This volume gives a rigorous account of the asymptotic theory of robust statistics, from the viewpoint of optimally robust functionals and their unbiased estimators and tests.

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