Bayes m-Truncated Sampling Allocations for Selecting the Best Bernoulli Population

Abstract

Let Р1,…,Рk be k ≥ 2 independent Bernoulli populations with success probabilities θ1,…,θk, respectively. Suppose we want to find the population with the largest success probability, using a Bayes selection procedure based on a prior density π(θ), which is the product of k known Beta densities, and a linear loss L(θ,i), θ= (θ1,…,θk), i = 1,…, k. Assume that k independent samples of sizes n 1,…,n k , respectively, have been observed already at a first stage, and that m more observations are planned to be taken at a future second stage. The problem considered is how to allocate these m observations in a suitable manner among the k populations, given the information gathered so far. Several allocation schemes, which have been considered previously, are compared analytically as well as numerically with the optimum allocation rule which is based on backward optimization. The numerical results indicate that there exists a good approximation to the optimum allocation rule which is easier to implement.