Extended Moment Series and the Parameters of the Negative Binomial Distribution

Recent studies indicate that, for finite sample sizes, moment estimators may be superior to maximum likelihood estimators in some regions of parameter space. In this paper a statistic based on the central moment of the sample is expanded in a Taylor series using 24 derivatives and many more terms than previous expansions. A summary algorithm is required to find meaningful approximants using the higher-order coefficients. A example is presented and a comparison between theoretical assessment and simulation results is made.