Continuity and differentiability of regression M functionals

MARÍA V. FASANO; RICARDO A. MARONNA; MARIELA SUED; VÍCTOR J. YOHAI

Valladolid

Conferencia; THE INTERNATIONAL CONFERENCE ON ROBUST STATISTICS; 2011

Universidad de Valladolid

This talk deals with the Fisher-consistency, weak continuity and differentiability of estimating functionals corresponding to a class of both linear and nonlinear regression high breakdown M estimates, which includes S and MM estimates. The usual types of differentiability, as Frechet or Hadamard differentiability, do not hold for regression M estimates. To overcome this problem, a restricted form of differentiability, called weak differentiability, is defined. This approach allows to prove the consistency and asymptotic normality of M estimates, including regression M estimates, under more general conditions than those required in standard approaches. In particular we prove that regression MM-estimates are asymptotically normal when the observations are phi-mixing.