In Computing Science and Statistics 24, (1992), pp. 592-596.

Optimal Allocation for Estimating the Product of Two Means

Janis Hardwick
Statistics Department, University of Michigan

Quentin F. Stout
EECS Department, University of Michigan

Abstract: Suppose we wish to estimate the product of the means of two independent populations of Bernoulli random variables, the parameters of which, themselves, are modeled as independent beta random variables. Assume that the total sample size for the experiment is fixed, but that the number of experimental units observed from each population may be random. Using a decision theoretic approach, we seek to minimize the Bayes risk that arises from using a squared error loss function.

Although selecting the form of an optimal estimator is critical to solving this problem, the real difficulty lies in determining an optimal strategy for sampling from the two populations. The problem of optimal estimation reduces, therefore, to a problem of optimal allocation which can be solved exactly using dynamic programming. We utilize similar programming techniques to evaluate exactly some of the other strategies that have been proposed for this problem. (The term exact is used repeatedly here to stress the fact that none of the computational results depend on simulation studies.)

Keywords: sequential allocation, nonlinear estimation, dynamic programming, reliability, myopic allocation, fixed allocation

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