Continuity and differentiability of regression M functionals

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

BERNOULLI - MATHEMATICAL STATISTICS AND PROBABILITY

INT STATISTICAL INST

Lugar: Amsterdam; Año: 2012 vol. 18 p. 1284 - 1309

1350-7265

This paper 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. A restricted type of differentiability, called weak differentiability, is defined, which suffices 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.