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Using Computational and Mathematical Methods to Explore a New Distribution: The $\nu$-Poisson

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

A new distribution (the $\nu$-Poisson) and its conjugate density are introduced and explored using computational and mathematical methods. The $\nu$-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 $\nu$-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 $\nu$-Poisson parameters is easily computed. We derive the necessary and sufficient condition for the conjugate family to be proper. The $\nu$-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 $\nu$-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.

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