Parameter estimation for generalized negative binomial distribution

The generalized negative binomial distribution (GNBD) has been found useful in the problems of random walk, queuing theory and branching processes. In this paper, GNBD parameters are estimated by the methods of (a) maximum likelihood, (a) first two moments and proportion of zeros, (c) first two moments and ratio of the first two frequencies, and (d) minimum chi-square estimation. The asymptotic relative efficiencies for the estimation methods are derived and compared. It is found that the method of minimum chi-square estimation is more efficient than the estimation methods in (b), and (c). From simulation results, the method in (b) seems to be the best when both bias and variance of the estimators are considered.