On the Bayes Estimate of a Functional of two Distributions

Abstract

Abstract

Let X₁, X₂ and Y_l. Y₂ be two indepently and identically distributed random variables each from the unknown distributions F and G respectively. Using a new constructive definition of the Dirichlet process we derive the Bayes estimate of the probability that {max (X₁, X₂) < min (Y₁, Y_z) or max (Y₁, Y₂) < min (X₁ X₂)}. We demonstrate how this constructive definition simplifies considerably the ‘linearized Dirichlet process’ approach adopted by Yamato (1975) to derive the Bayes estimate of a related functional. The limiting Bayes cstimato is shown to be asymptotically equivalent to a U-statistic, and its asymptotic normality is established. This problem relates to testing whether F and G arc identical.