A Koziol-Green Estimator for the conditional distribution function under dependent censoring

Abstract

In survival analysis, it is common to assume that the lifetime and the censoring time are independent. However in various data sets, we notice that this is not a correct situation and that the censoring time is informative for the estimation of the distribution of the lifetime. Koziol and Green develop a first informative censoring model in which they assume that the survival function of the censoring time is a power of the survival function of the lifetime. This model is afterwards generalized by Veraverbeke and Cardarso-Suarez (2000) to the fixed design regression setting. A second informative censoring model is developed by Zhang and Klein. They assume that the association between the lifetime and the censoring time is given by a known copula function. This model is further investigated by Rivest and Wells in the case of an Archimedean copula and by Braekers and Veraverbeke (2005) in a fixed design regression. In this paper, we combine both models. We assume on the one hand, a Archimedean copula for the association between the lifetime and the censoring time, and on the other hand that the conditional survival function of the censoring time is a function of the conditional survival function of the lifetime. In this model we find a nonparametric estimator for the conditional distribution function of the lifetime. As results, we prove an almost sure representation, uniform consistency and weak convergence for this estimator. Furthermore we construct an asymptotic confidence band for this estimator and compare it with the estimator of Braekers and Veraverbeke (2005).