Multistage Estimation Compared with Fixed-Sample-Size Estimation of the Negative Binomial Parameter k

Abstract

The negative binomial distribution often fits biological data well. Since the mean is of primary interest, the parameterization in terms of the mean and a parameter, k, has become almost standard. Given a precise estimate of k, sequential methods of estimating the mean are available. Precise estimation of k has been elusive. Estimators of k by the method of moments and by maximum likelihood estimation, based on samples of size 50 and 100, are investigated by using Monte Carlo methods. Although the maximum likelihood estimators tend to have smaller mean square errors than the method-of-moments estimators, neither can be considered good. A multistage approach to the estimation of k is presented, and Monte Carlo results indicate that this procedure gives a mean square error less than that of comparable fixed sample size estimators. Two truncation rules for the multistage procedure are also given and compared.