Allocation Schemes for Estimating the Product of Positive Parameters
Abstract Suppose that for i = 1, …, I, a random variable whose distribution depends on parameter μ i , > 0 is observable from population i. The problem is to estimate θ = IIμ i , using a Bayesian approach with squared error estimation loss in θ and allowing sequential allocation. That is, k observations may be taken one at a time and the decision to take an observation from population i may depend on past observations from all populations. For estimating θ, the best nonrandom allocation scheme, the myopic scheme, and the optimal allocation are discussed. The myopic scheme is typically asymptotically optimal and in several examples allocates in proportion to estimated population coefficients of variation, with the estimates updated at each stage. It is also shown that sequential allocation can improve the Bayes risk (over the best nonrandom allocation) up to 100 (I — 1)/I%.