Point estimators for the parameters of polya distribution of order k

We consider some issues of point estimation of the parameters of the Polya distribution of order k , which is a representative of the class of generalized Poisson distributions obtained with the geometrical distribution acting as the “generalizing” distribution. Important probability properties of this distribution are presented: expressions for the moments and a recurrence formula for the probability distribution function and its derivatives with respect to the parameters. These formulas are applied to compute the lower bound of the estimator covariance matrix. Two point estimation methods are explored: the method of moments and the substitution method (using the sample mean and the zero frequency). Asymptotic covariance matrices of these estimators are investigated, asymptotic efficiency of the estimation methods is computed, and some conclusions about the possibilities of the different estimators are presented.