Estimating a Product of Means: Bayesian Analysis with Reference Priors

Suppose that we observe X ∼ N(α, 1) and, independently, Y ∼ N(β, 1), and are concerned with inference (mainly estimation and confidence statements) about the product of means θ = αβ. This problem arises, most obviously, in situations of determining area based on measurements of length and width. It also arises in other practical contexts, however. For instance, in gypsy moth studies, the hatching rate of larvae per unit area can be estimated as the product of the mean of egg masses per unit area times the mean number of larvae hatching per egg mass. Approximately independent samples can be obtained for each mean (see Southwood 1978). Noninformative prior Bayesian approaches to the problem are considered, in particular the reference prior approach of Bernardo (1979). An appropriate reference prior for the problem is developed, and relatively easily implementable formulas for posterior moments (e.g., the posterior mean and variance) and credible sets are derived. Comparisons with alternative noninf...