  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.
