This thesis essentially takes two separate paths to solve the same problem, namely that of obtaining an estimator, a parameter of the negative binomial distribution, for which we can show that such properties as bias and variance of this estimator are "better" than corresponding properties of the simple moment estimator, the latter being the estimator which is used most often in practice. We first consider two moment estimators involving the third sample moment. In the case of both of these estimators, for a restricted range of the parameters and of sample size, these estimators are not an improvement over the simple moment estimator. In fact, for the range considered, the bias and variance of the simple moment estimator was always smaller. We then considered an estimator which was defined as the simple moment estimator for part of the sample space and defined as a constant elsewhere. This was primarily done to remove a "singularity" in the moment estimator. It was felt that this singularity was causing the large bias and variance which seemed to exist for certain values of the parameters. For n=100, the bias and variance were approximated in a range of interest of the parameters. The results indicate an improvement over the sample moment estimator. / Ph. D.