Sequentially Deciding between Two Experiments for Estimating a Common Success Probability

Abstract

To estimate a success probability p, two experiments are available: individual Bernoulli (p) trials or the product of τ individual Bernoulli (p) trials. This problem has its roots in reliability where either single components can be tested or a system of τ identical components can be tested. A total of N experiments can be performed, and the problem is to sequentially select some combination (allocation) of these two experiments, along with an estimator of p, to achieve low mean squared error (MSE) of the final estimate. This scenario is similar to that of the better-known group testing problem, but here the goal is to estimate failure rates rather than to identify defective units. The problem also arises in epidemiological applications such as estimating disease prevalence. Information maximization considerations, and analysis of the asymptotic MSE of several estimators, lead to the following adaptive procedure: use the maximum likelihood estimator (MLE) to estimate p, and if this estimator is below (above) the cutpoint aτ , then observe an individual (product) trial at the next stage. In a Bayesian setting with squared error estimation loss and suitable regularity conditions on the prior distribution, this adaptive procedure—replacing the MLE with the Bayes estimator—will be asymptotically Bayes. Exact computational evaluations of the adaptive procedure for fixed sample sizes show that it behaves roughly as the asymptotics predict. The exact analyses also show parameter regions for which the adaptive procedure achieves negative regret, as well as regions for which it achieves normalized MSE superior to that asymptotically possible. An example and a discussion of extensions conclude the work.