Optimal robust influence functions in semiparametric regression

Robust statistics allows the distribution of the observations to be any member of a suitable neighborhood about an ideal model distribution. In this paper, the ideal models are semi-parametric with finite-dimensional parameter of interest and a possibly infinite-dimensional nuisance parameter. In the asymptotic setup of shrinking neighborhoods, we derive and study the Hampel-type problem and the minmax MSE-problem. We show that, for all common types of neighborhood systems, the optimal influence function (psi) over tilde can be approximated by the optimal influence functions (psi) over tilde (n) for certain parametric models. For general semiparametric regression models, we determine ((psi) over tilde (n))(n is an element of N) in case of error-in-variables and in case of error-free-variables. Finally, the results are applied to Cox regression where we compare our approach to that of Bednarski [1993. Robust estimation in Cox's regression model. Scand. J. Statist. 20, 213-225] in a small simulation study and on a real data set. (C) 2009 Elsevier B.V. All rights reserved.