Sequential designs for estimating the product of paraeers

Abstract

Suppose that from each of I populations. an independent random variable with distribution depending on parameter can be observed. The goal is estimation of the product of population parameters, using a Bayesian approach and allowing sequential allocation and sampling. The Bayes estimator is used to estimate θ and thus, the problem involves two choices: the allocation procedure (where to take the observations) and the stopping rule (when to stop the experiment). Such a pair is called a policy, and policies are evaluated in terms of their expected total cost, here equal to the total Bayes risk.

For scaled squared error estimation loss. sufficient conditions for asymptotic optimality (a.o.) of a policy are derived, and a simple a.o.policy is construdted. For restricted population and prior distributions,the allocation proportions of a.o.policies are shown to be proportional to the population coefficient of variation and to not depend on g in the loss.Finally,the theory is applied to the reliability problem in which system is composed of I independent components connected in series,with μ the probability that the ith component works, and θ the probability that the system functions.

Keywords:

• allocation
• reliability
• Bayes estimators