Least-squares estimation of a convex discrete distribution

The least squares estimator of a discrete distribution under the constraint of convexity is introduced. Its existence and uniqueness are shown and consistency and rate of convergence are established. Moreover it is shown that it always outperforms the classical empirical estimator in terms of the Euclidean distance. Results are given both in the well- and the mis-specified cases. The performance of the estimator is checked throughout a simulation study. An algorithm, based on the support reduction algorithm, is provided. Application to the estimation of species abundance distribution is discussed.