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The histogram estimator of a discrete probability mass function oftenexhibits undesirable properties related to zero probability estimation bothwithin the observed range of counts and outside into the tails of thedistribution. To circumvent this, we formulate a novel second-order discretekernel smoother based on the recently developed mean-parametrizedConway--Maxwell--Poisson distribution which allows for both over- andunder-dispersion. Two automated bandwidth selection approaches, one based on asimple minimization of the Kullback--Leibler divergence and another based on amore computationally demanding cross-validation criterion, are introduced. Bothmethods exhibit excellent small- and large-sample performance. Computationalresults on simulated datasets from a range of target distributions illustratethe flexibility and accuracy of the proposed method compared to existingsmoothed and unsmoothed estimators. The method is applied to the modelling ofsomite counts in earthworms, and the number of development days of insect pestson the Hura tree.

Consistent second-order discrete kernel smoothing using dispersed Conway-Maxwell-Poisson kernels