Robust Asymptotic Statistics

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.

Klappentext

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.

Inhalt
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.- 2.4 Linear Regression.- 3: Asymptotic Statistics.- 3.1 General Remarks.- 3.2 Convolution Representation.- 3.3 Minimax Estimation.- 3.4 Testing.- 4: Nonparametric Statistics.- 4.1 Introduction.- 4.2 The Nonparametric Setup.- 4.3 Statistics of Functionals.- 4.4 Restricted Tangent Space.- 5: Optimal Influence Curves.- 5.1 Introduction.- 5.2 Minimax Risk.- 5.3 Oscillation.- 5.4 Robust Asymptotic Tests.- 5.5 Minimax Risk and Oscillation.- 6: Stable Constructions.- 6.1 The Construction Problem.- 6.2 M Equations.- 6.3 Minimum Distance.- 6.4 One-Steps.- 7: Robust Regression.- 7.1 The Ideal Model.- 7.2 Regression Neighborhoods.- 7.3 Conditional Bias.- 7.4 Optimal Influence Curves.- 7.5 Least Favorable Contamination Curves.- 7.6 Equivariance Under Basis Change.- 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 NeymanPearson 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.