Optimal transitive procedures for comparing large numbers of parameters

Abstract

Independent random samples are selected from each of a set of N independent populations, P1,…,Pn. Interest centers around comparing N (unknown) scalar parameters θ1,…,θN associated respectively with the N populations P1,…,Pn. Procedures are constructed for estimating the magnitude of each of the differences δt,j = θi − θj (1 ≤ i,j ≤ N) between pairs of populations. A loss function which adopts appropriate penalties for magnitude errors in estimation of differences is constructed. Magnitude estimators of differences are called transitive if they give rise to a transitive (i.e., consistent) relationship between pairwise differences of parameters. We show how to construct optimal effcient transitive magnitude–estimation procedures and demonstrate their usefulness through an example involving estimating the magnitude of the differences between disease incidence in paired towns for different pairs. Optimal transitive pairwise–comparison procedures are optimum (i.e., have the smallest posterior Bayes risks) in ...