A Weighted Poisson Distribution for Underdispersed Count Data

In this paper, we present a new weighted Poisson distribution for modeling underdispersed count data. Weighted Poisson distribution occurs naturally in contexts where the probability that a particular observation of Poisson variable enters the sample gets multiplied by some non-negative weight function. Suppose a realization y of Y a Poisson random variable enters the investigator’s record with probability proportional to w(y): Clearly, the recorded y is not an observation on Y, but on the random variable Yw, which is said to be the weighted version of Y. This distribution a two-parameter is from the exponential family, it includes and generalizes the Poisson distribution by weighting. It is a discrete distribution that is more flexible than other weighted Poisson distributions that have been proposed for modeling underdispersed count data, for example, the extended Poisson distribution (Dimitrov and Kolev, 2000). We present some moment properties and we estimate its parameters. One classical example is considered to compare the fits of this new distribution with the extended Poisson distribution.

EXPANSIONS FOR MOMENTS OF COMPOUND POISSON DISTRIBUTIONS

Expansions for moments of $\overline{X}$, the mean of a random sample of size n, are given for both the univariate and multivariate cases. The coefficients of these expansions are simply Bell polynomials. An application is given for the compound Poisson variable SN, where $S_{n} = n \overline{X}$ and N is a Poisson random variable independent of X1, X2, …, yielding expansions that are computationally more efficient than the Panjer recursion formula and Grubbström and Tang's formula.

On a transformation of the weighted compound Poisson process

We are using the following terminology—essentially following Feller:a) Compound Poisson VariableThis is a random variable where X1, X2, … Xn, … independent, identically distributed (X0 = o) and N a Poisson counting variablehence(common) distribution function of the Xj with j ≠ 0 or in the language of characteristic functionsb) Weighted Compound Poisson VariableThis is a random variable Z obtained from a class of Compound Poisson Variables by weighting over λ with a weight function S(λ)henceor in the language of characteristic functionsLet [Z(t); t ≥ o] be a homogeneous Weighted Compound Poisson Process. The characteristic function at the time epoch t reads thenIt is most remarkable that in many instances φt(u) can be represented as a (non weighted) Compound Poisson Variable. Our main result is given as a theorem.