Multiple monotonicity of discrete distributions: The case of the Poisson model

Abstract

Shape constrained estimation in discrete settings has received increasing attention in statistics. Among the most important shape constrained models is multiple monotonicity, including $k$-monotonicity, for a given integer $k\in [1,\infty )$, and complete monotonicity. Multiple monotonicity provides a nice generalization of monotonicity and convexity and has been successfully used in applications related to estimation of species richness. Although fully nonparametric, it is of great interest to determine some of the well-known parametric distributions which belong to this model. Among the most important examples are the family of Poisson distributions and mixtures thereof. In Giguelay (2017) $k$-monotonicity of Poisson distributions was connected to the roots of a certain polynomial, but a typographical error occurred while writing its expression. In this note, we correct that typographical error and give a detailed proof that a Poisson distribution with rate $\lambda \in [0,\infty )$ is $k$-monotone if and only if $\lambda \le \lambda _{k}$, where $\lambda _{k}$ is the smallest zero of the $k$-th degree Laguerre polynomial $L_{k}(x)=\sum _{j=0}^{k}(-1)^{j}\binom{k}{j}x^{j}/j!$, $x\ge 0$. This result yields the sufficient condition that a mixture of Poisson distributions is $k$-monotone if the support of the mixing distribution is included in $[0,\lambda _{k}]$. Furthermore, we show that the only complete monotone Poisson distribution is the Dirac distribution at $0$.