  In this paper a new generalization of the hyper-Poisson distribution is
proposed using the Mittag-Leffler function. The hyper-Poisson, displaced
Poisson, Poisson and geometric distributions among others are seen as
particular cases. This Mittag-Leffler function distribution (MLFD) belongs to
the generalized hypergeometric and generalized power series families and also
arises as weighted Poison distributions. MLFD is a flexible distribution with
varying shapes like non-increasing with unique mode at zero, unimodal with one
/ two non-zero modes. It can be under, equi or over dispersed. Various
distributional properties like recurrence relation for pmf, cumulative
distribution function, generating functions, formulae for different type of
moments, their recurrence relations, index of dispersion, its classification,
log-concavity, reliability properties like survival, increasing failure rate,
unimodality, and stochastic ordering with respect to hyper-Poisson distribution
have been discussed. The distribution has been found to fare well when compared
with the hyper-Poisson distributions in its suitability in empirical modeling
of differently dispersed count data. It is therefore expected that proposed
MLFD with its interesting features and flexibility, will be a useful addition
as a model for count data.
