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 ...