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 (X-sub-i, Y-sub-i) satisfying Y-sub-i > X-sub-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-sub-X(.) and a survival distribution S-sub-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-sub-X(.) or S-sub-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.