Allocation Proportional to Coefficients of Variation When Estimating the Product of Parameters

Abstract

Consider the problem of allocating a fixed total number of observations or a fixed budget to several populations with the goal of minimizing the variance of the product of sample means, where this product of sample means estimates the product of population means. The main application of this problem is to reliability, where the population mean is the Bernoulli success probability that the respective system component will function and the product is the probability that a series system composed of these components will function. The solution of the problem is shown to be allocation approximately proportional to population coefficients of variation or to the square root of the odds of failure in the reliability case. Furthermore, balanced allocation is shown to have a maximin property in terms of its asymptotic relative efficiency to optimal allocation. Finally, guidelines are given for constructing allocations that improve on balanced allocation.