  Non-parametric estimation of a convex discrete distribution may be of
interest in several applications, such as the estimation of species abundance
distribution in ecology. In this paper we study the least squares estimator of
a discrete distribution under the constraint of convexity. We show that this
estimator exists and is unique, and that it always outperforms the classical
empirical estimator in terms of the $\ell_{2}$-distance. We provide an
algorithm for its computation, based on the support reduction algorithm. We
compare its performance to those of the empirical estimator, on the basis of a
simulation study.
