Nonparametric location-scale models for censored successive survival times

Let (T₁,T₂) be gap times corresponding to two consecutive events, which are observed subject to (univariate) random right-censoring. The censoring variable corresponding to the second gap time T₂ will in general depend on this gap time. Suppose the vector (T₁,T₂) satisfies the nonparametric location-scale regression model T₂=m(T₁)+σ(T₁)?, where the functions m and σ are ‘smooth’, and ? is independent of T₁. The aim of this paper is twofold. First, we propose a nonparametric estimator of the distribution of the error variable under this model. This problem differs from others considered in the recent related literature in that the censoring acts not only on the response but also on the covariate, having no obvious solution. On the basis of the idea of transfer of tail information (Van Keilegom and Akritas, 1999), we then use the proposed estimator of the error distribution to introduce nonparametric estimators for important targets such as: (a) the conditional distribution of T₂ given T₁; (b) the bivariate distribution of the gap times; and (c) the so-called transition probabilities. The asymptotic properties of these estimators are obtained. We also illustrate through simulations, that the new estimators based on the location-scale model may behave much better than existing ones.