Estimating a Product of Means: Bayesian Analysis with Reference Priors

原文摘要

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 noninformative priors and with classical procedures are also given. The motivation for this work was in part the statistical importance of the problem and the difficulty in producing reasonable classical analyses, and in part to provide an interestingly complex example of a recently developed method of deriving reference priors for problems with nuisance parameters. This new method is briefly described. The problem is also of interest because of its mention by Efron (1986) as a situation for which standard noninformative prior Bayesian theories encounter difficulties.

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