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 semiparametric with finite-dimensional parameter of interest and a possibly infinite-dimensional nuisance parameter. 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 ψ ˜ can be approximated by the optimal influence functions ψ ˜ n for certain parametric models. neral semiparametric regression models, we determine ( ψ ˜ n ) n ∈ N in case of error-in-variables and in case of error-free-variables. y, the results are applied to Cox regression where we compare our approach to that of Bednarski [1993. Robust estimation in Coxʹs regression model. Scand. J. Statist. 20, 213–225] in a small simulation study and on a real data set.