Robust Differentiable Functionals in the Additive Hazards Model

Abstract

In this article we present a new family of estimators for the regression parameter β in the Additive Hazards Model which represents a gain in robustness not only against outliers but also against unspecific contamination schemes. They are consistent and asymptotically normal and furthermore, they have a nonzero breakdown point. In Survival Analysis the Additive Hazards Model proposes a hazard function of the form λ(t) = λ0(t) + β ′ z, where λ0(t) is a common nonparametric baseline hazard function and z is a vector of independent variables. For this model, the seminal work of Lin and Ying (1994) develops an estimator for the regression parameter β which is asymptotically normal and highly efficient. However, a potential drawback of that classical estimator is that it is very sensitive to outliers. In an attempt to gain robustness, Alvarez and Ferrarrio (2013) introduced a family of estimat ´ ors for β which are still highly efficient and asymptotically normal, but they also have bounded influence functions. Those estimators, which were developed using classical Counting Processes methodology, still retain the drawback of having a zero breakdown point. In this article we present a new family of estimators for the regression parameter β in the Additive Hazards Model which represents a gain in robustness not only against outliers but also against unspecific contamination schemes. They are consistent and asymptotically normal and furthermore, they have a nonzero breakdown point.