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 non parametric location-scale regression model T2=m(T1)+σ(T1)ɛ, where the functions m and σ are ‘smooth’, and ɛ is independent of T1. The aim of this paper is two fold. 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 non parametric 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 be have much better than existing ones.Ingrid Van Keilegom's research was financially supported by IAP research network P6/03 of the Belgian Government (Belgian Science Policy), and by the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement No. 203650. Jacob de Una-Alvarez acknowledges financial support from the project MTM2008-03129 of the Spanish Ministerio de Ciencia e Innovacion and also from the project PGIDIT07PXIB300191PR of the Xunta de Galicia. Luis F. Meira-Machado acknowledges financial support by Grant PTDC/MAT/104879/2008 (FEDER support included) of the Portuguese Ministry of Science, Technology and Higher Education and also from the project MTM2008-01603 of the Spanish Ministerio de Educacion y Ciencia