Robust Asymptotic Statistics, Softcover reprint of the original 1st ed. 1994
Volume I
Springer Series in Statistics Series

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.

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 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.