Non-parametric estimation and doubly-censored data: general ideas and applications to AIDS.

In many epidemiologic studies of human immunodeficiency virus (HIV) disease, interest focuses on the distribution of the length of the interval of time between two events. In many such cases, statistical estimation of properties of this distribution is complicated by the fact that observation of the times of both events is subject to intervalcensoring so that the length of time between the events is never observed exactly. Following DeGruttola and Lagakos, we call such data doubly-censored. Jewell, Malani and Vittinghoff showed that, with certain assumptions and for a particular doubly-censored data structure, non-parametric maximum likelihood estimation of the interval length distribution is equivalent to non-parametric estimation of a mixing distribution. Here, we extend these ideas to various other kinds of doubly-censored data. We consider application of the methods to various studies generated by investigations into the natural history of HIV disease with particular attention given to estimation of the distribution of time between infection of an individual (an index case) and transmission of HIV to their sexual partner.