Nonparametric estimation of the bivariate survival function with an application to vertically transmitted AIDS

In this paper we propose a nonparametric estimator of the bivariate survival function when the two durations are subject to right censoring. We consider the situation where the duration variables are successive and are always observed in a particular order, and the censoring mechanism bears on their sum, rather than on each separately. Thus, if they are correlated, the second duration is right censored by a dependent variable. A nonparametric maximum likelihood estimator is derived that takes the dependent censoring into account. We also derive the asymptotic variance of the estimator, and show how it can be estimated. The methods are applied on a vertically acquired, mother-to-child transmission, AIDS data set. The first duration corresponds to the HIV incubation period, and the second to the period of AIDS.