Estimation of the parameter in the discrete "Taxi" problem, with and without replacement

Abstract

In the continuous uniform distribution [0,N], the Maximum Likelihood estimator of N is known to possess high mean square error for large samples. This paper examines this issue in the discrete case, without replacement (extending the work of Tenenbein) and with replacement. Various other estimators including the Minimum Variance Unbiased estimator and Geary's closest estimator are compared in the continuous and two discrete cases. Recommendations are made for choice of estimator in each case, depending on the sample size and on imprecise information on N.