Some Properties of the Generalized Stuttering Poisson Distribution and Its Applications

Based on the probability generating function of stuttering Poisson distribution (SPD), this paper considers some equivalent propositions of SPD. From this, we show that some distributions in the application of non-life insurance actuarial science are SPD, such as negative binomial distribution, compound Poisson distribution etc.. By weakening condition of equivalent propositions of SPD, we define the generalized SPD and prove that any non-negative discrete random variable X with P{X = 0} > 0.5 obey generalized SPD. Then, we discuss the waiting time distribution of generalized stuttering Poisson process. We consider cumulant estimation of generalized SPD's parameters. As an application, we use SPD with four parameters (4th SPD) to fit auto insurance claim data. The fitting results show that 4th SPD is more accurate than negative binomial and Poisson distribution.