Two stage design for estimating the product of means with cost in the case of the exponential family

TL;DR

This paper proposes a two-stage Bayesian design for estimating the product of means from exponential family populations, optimizing the allocation of observations to minimize Bayes risk under cost constraints, and proves its asymptotic optimality.

Contribution

It introduces a novel two-stage sampling design that minimizes Bayes risk for product of means estimation in exponential families, with proven asymptotic optimality.

Findings

The design minimizes Bayes risk under cost constraints.

Asymptotic optimality of the proposed design is established.

Efficient allocation of observations improves estimation accuracy.

Abstract

We investigate the problem of estimating the product of means of independent populations from the one parameter exponential family in a Bayesian framework. We give a random design which allocates mi the number of observations from population Pi such that the Bayes risk associated with squared error loss and cost per unit observation is as small as possible. The design is shown to be asymptotically optimal.