ON HADAMARD DIFFERENTIABILITY AND M-ESTIMATION IN LINEAR MODELS by

Robust (M-) estimation in linear models generally involves statistical functional processes. For drawing statistical conclusions (in large samples), some (uniform) linear approximations are usually needed for such functionals. In this context, the role of Hadamard differentiability is critically examined in this dissertation. In particular, the concept of the second-order Hadamard differenti-ability and some related theoretical results are established for the study of the convergence rate in probability of the uniform asymptotic linearity of the M-estimator of regression. Thereby, using Hadamard differentiability through the linear approximation of the estimator, the asymptotic normality, the weak consistency and an asymptotic representation are derived under the weak conditions on the score function t/J, the underlying d.f. F, and the regreSSIOn constants. Some other approaches are also considered for the study of the asymptotic normality and the weak consistency, and these approaches lead to two sets of conditions for the strong consistency of the M-estimators in linear models.