Robust Asymptotic Statistics: Volume I

ISBN-13: 9781468406269 / Angielski / Miękka / 2012 / 390 str.

Helmut Rieder

Opis

1 To the king, my lord, from your servant Balasi: 2 ... The king should have a look. Maybe the scribe who reads to the king did not understand . . . . shall I personally show, with this tablet that I am sending to the king, my lord, how the omen was written. 3 Really, he who has not followed the text with his finger cannot possibly understand it. This book is about optimally robust functionals and their unbiased esti mators and tests. Functionals extend the parameter of the assumed ideal center model to neighborhoods of this model that contain the actual distri bution. The two principal questions are (F): Which functional to choose? and (P): Which statistical procedure to use for the selected functional? Using a local asymptotic framework, we deal with both problems by linking up nonparametric statistical optimality with infinitesimal robust ness criteria. Thus, seemingly separate developments in robust statistics are presented in a unifying way."

Spis treści

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.