## Hazard Functions of Exponential Mixtures and Their Link With Mixed Poisson Distributions

##### Abstract

In this study the mixed Poisson distribution has been defined in terms of the hazard
function of an exponential mixture. This work, therefore, has shown that there is a
link between exponential and Poisson mixtures, such that a hazard function of an
exponential mixture characterizes an infinitely divisible mixed Poisson distribution,
which is also a compound Poisson distribution.
It has been established that since a hazard function of an exponential mixture
is completely monotone, then the mixing distribution is infinitely divisible through
Laplace transform; and a Poisson mixture with an infinitely divisible mixing distribution
is infinitely divisible too. Further, an infinitely divisible mixed Poisson
distribution is a compound Poisson distribution. The compound Poisson distribution
has been constructed recursively in terms of the probability mass function
(pmf) of the independent and identically distributed (i.i.d.) random variables and
the hazard function of the exponential mixture. It has also been shown that a
sum of hazard functions of exponential mixtures gives rise to a convolution of
infinitely divisible Poisson mixtures, hence a convolution of compound Poisson
distributions. Given the importance of hazard functions of exponential mixtures
in the development of these models, the hazard functions have been constructed
using continuous mixing distributions through probability density functions and
survival functions, and using Laplace transforms of probability density functions
in continuous compound distributions.
From the literature reviewed, it was found that there are mixing distributions
that have been used in the construction of mixed Poisson distributions that are not
part of the exponential mixtures literature. Type I and type II exponential mixtures,
that are in explicit form, in terms of modified Bessel function of the third kind and
in terms of confluent hyper-geometric function, have been constructed using these
mixing distributions. Whereas hazard functions of some of the exponential mixtures
constructed are single hazard functions, others are sums of hazard functions.
However, the Mellin transform technique that was used to obtain moments failed in
some cases and this necessitated the use of an alternative method, the conditional
expectation technique. The models developed were applied to a class of mixed
Poisson distribution known as Hofmann distributions to show the link between
exponential and Poisson mixtures.
The tools or methodologies in this study include; special functions, which
have been used to construct some mixing distributions and exponential mixtures;
transformations, which have been used to obtain moments of the mixtures that
are in terms of the modified Bessel function of the third kind and confluent hypergeometric
function; generating functions, which have been used to determine the
corresponding mixed Poisson distribution and the pmf of the iid random variables;
conditional expectation, which has been used as an alternative technique in cases
where the Mellin transform fails.
Although the Hofmann hazard function is a good illustration of the theory,
1
there is need to consider other classes of hazard functions, particularly those based
on frailty models. There is room for further research to identify other families of
hazard functions of exponential mixtures, which are not necessarily members of
the family of Hofmann distributions, and whose sums of hazard functions give rise
to convolutions of Poisson mixtures.

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