Single index regression models in the presence of censoring depending on the covariates

Consider a random vector $(X',Y)'$, where $X$ is $d$-dimensional and $Y$ is one-dimensional. We assume that $Y$ is subject to random right censoring. The aim of this paper is twofold. First we propose a new estimator of the joint distribution of $(X',Y)'$. This estimator overcomes the common curse-of-dimensionality problem, by using a new dimension reduction technique. Second we assume that the relation between $X$ and $Y$ is given by a single index model, and propose a new estimator of the parameters in this model. The asymptotic properties of all proposed estimators are obtained.