Nearly second order three-stage design for estimating a product of several Bernoulli proportions

We give a second order lower bound for the variance incurred by a three-stage procedure for estimating a product of means by allocation from independent Bernoulli populations. The asymptotic analysis is derived from the trivial case of two proportions which allows one to construct an exact policy with an exact lower bound. We extend the result to the case of several proportions and rigorously prove the nearly second order asymptotic optimality of the proposed scheme using the rate of convergence in the strong law of large numbers which is upper limited by the central limit theorem. The results are validated via Monte-Carlo simulations and are very promising for the exact second order optimality.