Many chronic diseases, including AIDS and cancer, do not manifest themselves clinically until some time after their inception. In studies of disease natural history, the duration of the asymptomatic period is of interest-in AIDS, to predict the epidemic's course, and in cancer, to develop efficient screening strategies. This article provides a bridge between the two fields with respect to estimation of the asymptomatic period. By adapting AIDS methodology to cancer, the article identifies a non-parametric method for estimating the duration of the asymptomatic period in cancer. The method is similar to one developed by Louis et al. (Mathematical Biosciences, 40, 111-144 (1978)), and is designed to apply to data from a cohort of individuals, screened periodically. After reviewing the similarities and differences between the AIDS and cancer contexts, we develop an EM algorithm that, at convergence, yields a maximum or saddle point of the likelihood. We investigate the performance of the algorithm by means of a simulation study, explore the effect of adding a smoothing step to the estimation procedure, and adapt the method for use with a data set in which disease prevalence is low. We apply the method to data from the HIP breast cancer screening trial.