Inference for the binomial N parameter: A hierarchical Bayes approach

SUMMARY A hierarchical Bayes approach to the problem of estimating N in the binomial distribution is presented. This provides a simple and flexible way of specifying prior information, and also allows a convenient representation of vague prior knowledge. It yields solutions to the problems of interval estimation, prediction and decision making, as well as that of point estimation. The Bayes estimator compares favourably with the best, previously proposed, point estimators in the literature. unknown parameters N and 0. Most of the literature about statistical analysis of this model has focused on point estimation of N, while interval estimation, prediction and decision making have been little considered; see ? 2. I adopt a hierarchical Bayes approach. This provides a simple way of specifying prior information, and also allows a convenient representation of vague prior knowledge using limiting, improper, prior forms. It leads to solutions of the problems of interval estimation, prediction and decision making, as well as that of point estimation. A difficulty with Bayesian analysis of this problem has been the absence of a sufficiently flexible and tractable family of prior distributions, mainly due to the fact that N is an integer. The present approach gets around this by first assuming that N has a Poisson distribution. The resulting hyperparameters are then continuous-valued, and one may use existing results about conjugate and vague priors in better understood settings.