Estimation of a convex discrete distribution

TL;DR

This paper introduces a unique least squares estimator for convex discrete distributions, demonstrating its superiority over the empirical estimator in accuracy and providing an efficient algorithm for its computation.

Contribution

The paper develops a novel least squares estimator for convex discrete distributions, proving its existence, uniqueness, and improved performance over traditional methods.

Findings

Estimator always outperforms empirical in $\, ext{l}_2$-distance

Provides an algorithm based on support reduction for computation

Simulation studies confirm improved accuracy

Abstract

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.