Fitting Negative Binomial Distributions by the Method of Maximum Likelihood

Maximum likelihood solutions for negative binomial distributions have been worked out by a number of authors. The purpose of this paper will be to develop the solutions in an insurance context, and to investigate one phase that has not been touched upon in the literature. The formulas developed will be applied to some actual data. Dropkin 1 has considered the process of fitting the negative binomial distribution by the method of moments, to a set of complete data. In this paper, the same problem will be treated using the method of maximum likelihood; but first, two problems will be solved where the number of observations in the zero case (claim-free insureds in insurance applications) is subject to some special condition.