We use a penalized likelihood approach to obtain a smooth estimate of a bivariate distribution from grouped data where each observation consists of a region in a plane. The purpose of the analysis is to estimate the incubation period distribution of AIDS from the Multicenter AIDS Cohort Study, a prevalent cohort of homosexual men. In this article we illustrate the usefulness of the penalized likelihood approach. We also discuss the use of a cross-validation and a Bayesian scheme to choose the smoothing parameters and bootstrap samples to assess uncertainty.