Point estimates of parameters for Neuman distribution of order k

We construct point estimates of the parameters of a Neuman distribution of order k as a representative of the class of generalized Poisson distributions. The main properties of this distribution (a recurrence formula, cumulants and moments, derivatives with respect to parameters) are given in a system with infinitely many parameters, and the relationships are demonstrated with the previously obtained expressions in a two-parameter system. Among the point estimation methods we consider the moment method and the substitution method, which both lead to simple systems of equations; the solvability conditions for these systems are investigated. The efficiency of the estimators relative to the Cramer-Rao lower bound is examined and some conclusions are drawn regarding their applicability. The equations of the maximum likelihood estimation method are written out for infinitely many parameters and for the two-parameter case. __________ Translated from Prikladnaya Matematika i Informatika, No. 28, pp. 93–109, 2008.