Robust Asymptotic Statistics

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.