Censoring heavy-tail count distributions for parameters estimation with an application to stable distributions

Some families of count distributions do not have a closed form of the probability mass function and/or finite moments and therefore parameter estimation can not be performed with the classical methods. When the probability generating function of the distribution is available, a new approach based on censoring and moment criterion is introduced, where the original distribution is replaced with that censored by using a Geometric distribution. Consistency and asymptotic normality of the resulting estimators are proven under suitable conditions. The crucial issue of selecting the censoring parameter is addressed by means of a data-driven procedure. Finally, this novel approach is applied to the discrete stable family and the finite sample performance of the estimators is assessed by means of a Monte Carlo simulation study.