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**Using Computational and Mathematical Methods to Explore a New
Distribution: The -Poisson**

**Galit Shmueli, Thomas P. Minka, Joseph B. Kadane, Sharad Borle
and Peter Boatwright**

A new distribution (the -Poisson) and its conjugate
density are introduced and explored using computational and
mathematical methods. The -Poisson is a two-parameter
extension of the Poisson distribution that generalizes some
well-known discrete distributions (Poisson, Bernoulli, Geometric).
It also leads to the generalization of distributions derived from
these discrete distributions (viz. the Binomial and Negative
Binomial). We use mathematics as far as we can and then employ
computational and graphical methods to explore the distribution
and its conjugate density further. Three methods are presented for
estimating the -Poisson parameters: The first is a fast
simple weighted least squares method, which leads to estimates
that are sufficiently accurate for practical purposes. The second
method of maximum likelihood can be used to refine the initial
estimates. This method requires iterations and is more
computationally intensive. The third estimation method is
Bayesian. Using the conjugate prior, the posterior density of the
-Poisson parameters is easily computed. We derive the
necessary and sufficient condition for the conjugate family to be
proper. The -Poisson is a flexible distribution that can
account for over/under dispersion commonly encountered in count
data. We also explore an empirical application demonstrating this
flexibility of the -Poisson to fit count data which does not
seem to follow the Poisson distribution.
*Keywords:* Poisson Distribution; Bernoulli
Distribution; Geometric Distribution; Elicitation; Conjugate
Family; Exponential Family.

*Heidi Sestrich*

*6/12/2001*
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