Estimation in nonparametric location-scale regression models with censored data

Abstract

Consider the random vector (X, Y), where X is completely observed and Y is subject to random right censoring. It is well known that the completely nonparametric kernel estimator of the conditional distribution ${F(\cdot|x)}$ of Y given X = x suffers from inconsistency problems in the right tail (Beran 1981, Technical Report, University of California, Berkeley), and hence any location function m(x) that involves the right tail of ${F(\cdot|x)}$ (like the conditional mean) cannot be estimated consistently in a completely nonparametric way. In this paper, we propose an alternative estimator of m(x), that, under certain conditions, does not share the above inconsistency problems. The estimator is constructed under the model Y = m(X) + σ(X)ε, where ${\sigma(\cdot)}$ is an unknown scale function and ε (with location zero and scale one) is independent of X. We obtain the asymptotic properties of the proposed estimator of m(x), we compare it with the completely nonparametric estimator via simulations and apply it to a study of quasars in astronomy.