Optimal Allocation for Estimating the Mean of a Bivariate Polynomial

Janis Hardwick      Quentin F. Stout
University of Michigan


Abstract: Suppose we wish to estimate the mean of some polynomial function of random variables from two independent Bernoulli populations, the parameters of which, themselves, are modeled as independent beta random variables. It is assumed that the total sample size for the experiment is fixed, but that the number of experimental units observed from each population may be random. This problem arises, for example, when estimating the fault tolerance of a system by testing its components individually.

The goal is 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. We solve the allocation problem via dynamic programming. Similar programming techniques are utilized to give exact evaluation of properties of a number of ad hoc allocation strategies that might also be considered for use in this problem. Two sample polynomials are analyzed along with a number of examples indicating the effects of different prior parameters settings. The effects of differences between prior parameters used in the design and analysis stages of the experiment are also examined.

Keywords: nonlinear estimation, sequential design, robustness, response adaptive sampling, design of experiments, myopic, hyperopic, dynamic programming, adaptive allocation, product, Bayesian, fault tolerance, stochastic optimization, path induction

Complete paper. This paper appears in Sequential Analysis 15 (1996), 71-90.

This extends earlier work appearing in J. Hardwick and Q.F. Stout, "Optimal allocation for estimating the product of two means", Computing Science and Statistics 24 (1992), pp. 597-601.


Related Work
Adaptive Allocation:
Here is an explanation of this topic, and here are our relevant publications.
Dynamic Programming (also known as backward induction):
Here is an overview of our work.

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