  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. We consider cumulant
estimation of generalized SPD's parameters. As an application, we use SPD with
four parameters (4-th SPD) to fit auto insurance claim data. The fitting
results show that 4-th SPD is more accurate than negative binomial and Poisson
distribution.
