Estimating survival under a dependent truncation

Abstract

The product-limit estimator calculated from data subject to random left-truncation relies on the testable assumption of quasi-independence between the failure time and the truncation time. In this paper, we propose a model for a truncated sample of pairs ( X i , Y i ) satisfying Y i > X i . A possible dependency between the truncation time and the variable of interest is modelled with a parametric family of copulas. The model also features a distribution function F X (.) and a survival distribution S Y (.) associated with the marginal behaviours of X and Y in the observable region Y > X . Semiparametric estimators for these two functions are proposed; they do not make any parametric assumption about either F X (.) or S Y (.). We derive an estimator for the copula parameter α based on the conditional Kendall's tau. We generalise the copula-graphic estimators of Zheng & Klein (1995) to truncated variables. The asymptotic distributions of all these estimators are then investigated. The methods are illustrated with a real dataset on HIV infection by transfusion and by simulations.