Estimating survival under a dependent truncation.

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 (Xi,Yi) satisfying Yi > Xi. 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 FX(.) and a survival distribution SY(.) 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 FX(.) or SY(.). 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. [ABSTRACT FROM AUTHOR]