Can We Estimate N?
Feb 29 1992
Estimating N in a binomial problem is notoriously difficult. It is well known that the posterior mean of N is highly unstable. We show that with arbitrarily small contaminations of the prior the posterior mean is arbitrarily large. Some authors turn to the median of the posterior as a more stable estimator. We show, however, that with contaminations to the prior above a critical value, the posterior median can be arbitrarily large. In our examples, this critical value can be quite small even if the prior is proper. We provide an upper bound on this critical value over all priors. We also provide results on posterior medians and modes for density ratio classes of priors. Some authors study the sensitivity to perturbations of the data, tacitly assuming that the maximum data point is the most influential point. We show that this is false and that, in fact, the minimum may be more influential.
Keywords:N-estimation, Robust Bayesian inference, sensitivity analysis