Continuity and differentiability of regression M functionals

FASANO, V.; MARONNA, R.A.; SUED, M.; YOHAI, V. J.

BERNOULLI - MATHEMATICAL STATISTICS AND PROBABILITY

INT STATISTICAL INST

Lugar: Amsterdam; Año: 2012 p. 1284 - 1309

1350-7265

This paper deals with the Fisher–consistency, weak continuity and differentiability of estimatingfunctionals corresponding to a class of both linear and nonlinear regression high breakdown M estimates, which includes S and MM estimates. A restricted type of differentiability, called weak di¤erentiability, is de…ned, which succes to prove the asymptotic normality of estimates based on the functionals. This approach allows to prove the consistency, asymptotic normality and qualitative robustness of 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.