In some epidemiologic studies, identification of individuals for study is dependent on the occurrence of some event. Once an individual is identified, the time of a previous event, termed an initiating event, is determined retrospectively. This article considers problems of estimation when initiating events occur as a nonhomogeneous Poisson process, and the time s from the initiating event to the final event has pdf f(s) independent of the time of the initiating event. A simple form for the likelihood function is obtained and methods of parametric and nonparametric estimation are developed and considered. In particular, the model is related to a Poisson process in the plane, and for the parametric case simple algorithms are developed for parameter estimation. Regression models are also considered as well as various generalizations of the basic problem. Parallel to the theoretical development, data on patients diagnosed with acquired immune deficiency syndrome (AIDS) are considered and a detailed analysis is given. The data report the dates of diagnosis with AIDS and infection with human immunodeficiency virus, for patients reported to the Centers of Disease Control in Atlanta, Georgia, and thought to be infected by blood or blood-product transfusion. The analysis of these data was considered by Medley, Anderson, Cox, and Billard (1987), Lui et al. (1986), and others. It is shown that nonparametric analysis leads to simple estimates of certain parameters and indicates clearly the nature of an identifiability problem that arises with data of this kind. Problems arise in the estimation of the total number of infectives or percentiles of the distribution of the induction period, s.