A Weighted Poisson Distribution for Underdispersed Count Data

Abstract

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.

On Underdispersed Count Kernels for Smoothing Probability Mass Functions

“…We note that any underdispersed count distribution cannot always lead to its corresponding associated kernel; see Appendix A and also [23,24]. It would also be better to improve the bandwidth selection with data-driven methods; Appendix B mentions the direction for the local Bayesian bandwidth selection.…”