Nonparametric Estimation of the Bivariate Survival Function with Truncated Data

Randomly left or right truncated observations occur when one is concerned with estimation of the distribution of time between two events and when one only observes the time if one of the two events falls in a fixed time-window, so that longer survival times have higher probability to be part of the sample than short survival times. In AIDS applications the time between seroconversion and AIDS is only observed if the person did not die before the start of the time-window. Hence, here the time of interest is truncated if another related time-variable is truncated. This problem is a special case of estimation of the bivariate survival function based on truncation by a bivariate truncation time, the problem covered in this paper; in the AIDS application one component of the bivariate truncation time-vector is always zero. In this application the bivariate survival function is of interest itself in order to study the relation between time until AIDS and time between AIDS and death. We provide a quick algorithm for computation of the NPMLE. In particular, it is shown that the NPMLE is explicit for the special case when one of the truncation times is zero, as in the AIDS application above. We prove that the NPMLE is consistent under a minimal condition. Moreover, we prove asymptotic normality under a tail assumption at the origin. The condition holds in particular if the truncation distribution has an atom at zero. We provide an algorithm for estimation of its limiting variance. By simply plugging in one of the several proposals for estimation of the bivariate survival function based on right-censored data in the estimating equation we obtain an estimator based on right-censored randomly truncated data. Here, substitution of an estimator which handles the right-censoring efficiently leads to an efficient estimator