In this paper we make an attempt to review count data models developed so far as generalizations of Poisson process. We consider Winkleman’s gamma count model and the Weibull count model of Mc Shane et al. The fractional generalization of Poisson process by Mainardi et al. is also considered. A Mittag-Leffler count model is developed and studied in detail. Simulation studies are also conducted.

Using the Hamilton-Jacobi-Bellman equation, we derive both a Keynes-Ramsey rule and a closed form solution for an optimal consumption-investment problem with labor income. The utility function is unbounded and uncertainty stems from a Poisson process. Our results can be derived because of the proofs presented in the accompanying paper by Sennewald (2006). Additional examples are given which hig...

Our purpose in this paper is to apply the general methodology for model selection based on T-estimators developed in Birgé (2006a) to the particular situation of the estimation of the unknown mean measure of a Poisson process. We introduce a Hellinger type distance between finite positive measures to serve as our loss function and we build suitable tests between balls (with respect to this dist...

The expected value of a functional F(η) of a Poisson process η can be considered a function of its intensity measure μ . The paper surveys several results concerning differentiability properties of this functional on the space of signed measures with finite total variation. Then necessary conditions for μ being a local minima of the considered functional are elaborated taking into account possi...

The Poisson process is a stochastic counting process that arises naturally in a large variety of daily-life situations. We present a few definitions of the Poisson process and discuss several properties as well as relations to some well-known probability distributions. We further briefly discuss the compound Poisson process.