Abstract: We develop an estimation procedure for a discrete probability mass function (pmf) with unknown support. We derive its maximum likelihood estimator under the mild and natural shape-constraint of unimodality. Shape-constrained estimation is a powerful and robust technique that additionally provides smoothing of the empirical distribution yielding gains in efficiency. We show that our unimodal estimator is consistent when the model is specified, and that it converges to the best projection of the true pmf on the unimodal class under model misspecification. We derive the limiting distribution of the the estimator, and use this to build asymptotic confidence bands for the unknown pmf when the latter is unimodal. We illustrate our approach using time-to-onset data of the Ebola virus during the 1976 outbreak in the former republic of Zaire.

Key words and phrases: maximum likelihood estimation, probability mass function estimation, shape constrained estimation, unimodal.