The Canadian Journal of Statistics Vol. 41, No. 3, 2013, Pages 439–452 La revue canadienne de statistique 439 Nonparametric estimation of the conditional survival function for bivariate failure times Lajmi LAKHAL-CHAIEB 1 *, Belkacem ABDOUS 2 and Thierry DUCHESNE 1 1 D´ epartement de math´ ematiques et statistique, Universit´ e Laval, Qu´ ebec, Canada G1V 0A6 2 D´ epartement de m´ edecine sociale et pr´ eventive, Universit´ e Laval, Qu´ ebec, Canada G1V 0A6 Key words and phrases: Bandwidth selection; gap times; generalized Kaplan–Meier; local polynomial; multivariate survival; semi-competing risks; successive failure times. MSC 2010: Primary 62G08; secondary 62N02 Abstract: This paper discusses the nonparametric estimation of the conditional survival function for bivari- ate failure time data in the presence of dependent censoring. We use local polynomial smoothing techniques to obtain a simple estimator of the conditional survival function and investigate its asymptotic proper- ties. Several censoring structures are considered. The proposal is compared to an existing alternative by simulation and illustrated with two real data sets. The Canadian Journal of Statistics 41: 439–452; 2013 © 2013 Statistical Society of Canada R´ esum´ e: Cet article porte sur l’estimation non param´ etrique de la fonction de survie conditionnelle pour des donn´ ees bivari´ ees de dur´ ees de vie en pr´ esence de censure d´ ependante. Les auteurs utilisent des techniques de lissage par polynˆ omes locaux pour obtenir un estimateur simple de la fonction de survie conditionnelle et ils ´ etudient ses propri´ et´ es asymptotiques. Plusieurs structures de censure sont examin´ ees. Les auteurs comparent leur m´ ethode ` a une technique existante au moyen d’une simulation et ils l’illustrent ` a l’aide de deux jeux de donn´ ees r´ eelles. La revue canadienne de statistique 41: 439–452; 2013 © 2013 Société statistique du Canada 1. INTRODUCTION Multivariate failure time data arise when patients in an experiment or a follow-up study are potentially subject to several related events. The association between the occurrence times of these events is often of prime interest to practitioners and may be depicted through the survival function of the time of occurrence of an event, say T 2 , given the occurrence time of another event, say T 1 . Inference about such a conditional survival function constitutes the main purpose of numerous scientific studies. For instance Nan et al. (2005, 2006) estimate the survival function associated to the age at onset of menopause given the age of onset of some bleeding event in a woman’s reproductive life, and Xu, Kalbfleisch, & Tai (2010) and Lakhal-Chaieb, Cook, & Lin (2010) estimate the survival function of time until death given the age at onset of nasopharyngeal and colon cancer recurrence, respectively. While the aforementioned authors proposed semi- parametric models, in this paper we are concerned with the completely nonparametric estimation of the conditional survival function π 2|1 (t 2 |t 1 ) = Pr(T 2 >t 2 |T 1 = t 1 ). Additional supporting information may be found in the online version of this article at the publisher’s web-site. * Author to whom correspondence may be addressed. E-mail: lakhal@mat.ulaval.ca © 2013 Statistical Society of Canada / Société statistique du Canada