Nonparametric Location-Scale Models for Censored Successive Survival Times Ingrid Van Keilegom ∗ Jacobo de U ˜ na- ´ Alvarez † Luis Meira-Machado ‡ September 9, 2010 Abstract Let (T 1 ,T 2 ) be gap times corresponding to two consecutive events, which are ob- served subject to (univariate) random right-censoring. The censoring variable cor- responding to the second gap time T 2 will in general depend on this gap time. Sup- pose the vector (T 1 ,T 2 ) satisfies the nonparametric location-scale regression model T 2 = m(T 1 )+ σ(T 1 )ε, where the functions m and σ are ‘smooth’, and ε is indepen- dent of T 1 . 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 distri- bution of T 2 given T 1 ; (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. Key words: Bivariate distribution; Conditional distribution; Error distribution; Progres- sive three-state model; Recurrent events; Transfer of tail information; Transition proba- bilities. * Institute of Statistics, Universit´ e catholique de Louvain, Voie du Roman Pays 20, 1348 Louvain-la- Neuve, Belgium, E-mail: ingrid.vankeilegom@uclouvain.be † Departamento de Estad´ ıstica e Investigaci´ on Operativa, Facultad de CC. Econ´omicas y Empresar- iales, University of Vigo, Campus Universitario Lagoas-Marcosende, 36310 Vigo, Spain, E-mail: ja- cobo@uvigo.es ‡ Department of Mathematics and Applications, University of Minho, Guimar˜ aes, Portugal, E-mail: lmachado@math.uminho.pt 1