Robust estimation in single index models when the errors have a unimodal density with unknown nuisance parameter

In this paper, we propose a robust profile estimation method for the parametric and nonparametric components of a single index model when the errors have a strongly unimodal density with unknown nuisance parameter. Under regularity conditions, we derive consistency results for the link function estimators as well as consistency and asymptotic distribution results for the single index parameter estimators. Under a log--Gamma model, the sensitivity to anomalous observations is studied by means of the empirical influence curve. We also discuss a robust $K-$fold procedure to select the smoothing parameters involved. A numerical study is conducted to evaluate the small sample performance of the robust proposal with that of their classical relatives, both for errors following a log--Gamma model and for contaminated schemes. The numerical experiment shows the good robustness properties of the proposed estimators and the advantages of considering a robust approach instead of the classical one.