The histogram estimator of a discrete probability mass function often exhibits undesirable properties related to zero probability estimation both within the observed range of counts and outside into the tails of the distribution. To circumvent this, we formulate a novel second-order discrete kernel smoother based on the recently developed mean-parametrized Conway--Maxwell--Poisson distribution which allows for both over- and under-dispersion. Two automated bandwidth selection approaches, one based on a simple minimization of the Kullback--Leibler divergence and another based on a more computationally demanding cross-validation criterion, are introduced. Both methods exhibit excellent small- and large-sample performance. Computational results on simulated datasets from a range of target distributions illustrate the flexibility and accuracy of the proposed method compared to existing smoothed and unsmoothed estimators. The method is applied to the modelling of somite counts in earthworms, and the number of development days of insect pests on the Hura tree.

翻译：离散概率质量函数的直方图估计器往往显示出与所观察到的数数范围内和外向分布尾部的零概率估计值有关的不良特性。为绕过这一特性,我们根据最近开发的允许超散和低散分布的近似平衡-马克斯韦尔-波西松分布,制作了一个新的第二阶离散内核滑动器。两种自动带宽选择方法,一种基于简单最小化库尔回背-利伯尔差异,另一种基于更具有计算要求的交叉校验标准。两种方法都表现出极好的小型和大型模量性性性能。从一系列目标分布中模拟数据集的计算结果表明拟议方法与现有光滑和无吸附的估测器相比的灵活性和准确性。该方法适用于土虫中索地点计数的建模,以及胡拉树上的昆虫生长日数。