Nonparametric maximum likelihood estimation for bivariate censored data

We study the behavior of the (nonparametric) maximum likelihood estimator (MLE) for bivariate censored data. The motivation for doing this was triggered by our interest in the problem of estimating the incubation time distribution of HIV/AIDS. We study the computational and algorithmic aspects of the MLE for bivariate interval censored data, and introduce an algorithm for computing the MLE which is significantly faster than the other methods of which we are aware. The main improvement in the speed of the algorithm is due to a great simplification of the preliminary search for regions where the MLE can have positive mass, which is essentially a parameter reduction method. Furthermore, we study theoretical properties of the MLE for models connected with the problem of estimating the incubation time distribution of HIV/AIDS. In particular, we give an interpretation of its definition in terms of graph theory and discuss different types of non-uniqueness that can arise. We show that the naive MLE is inconsistent in two of the models that we consider. We introduce other estimators which can be viewed as modifications or extensions of the idea of maximum likelihood, and which, under certain conditions on the underlying distributions, will be consistent. Our methods are illustrated by an analysis of data from the Amsterdam Cohort Study on injecting drug users and we construct several visualizations of the data. We point out that a fundamental problem in the estimation of the incubation time distribution is the fact that, without further assumptions on the underlying distributions which allow some kind of extrapolation, we can only estimate the bivariate distribution consistently in a fixed rectangular region of the plane.