Asymptotic Solutions of the Compound Decision Problem for Two Completely Specified Distributions

Abstract

A compound decision problem consists of the simultaneous consideration of $n$ decision problems having identical formal structure. Decision functions are allowed to depend on the data from all $n$ components. The risk is taken to be the average of the resulting risks in the component problems. A heuristic argument for the existence of good asymptotic solutions was given by Robbins ([1] Sec. 6) and was preceded by an example (component decisions between $N(-1,1)$ and $N(1,1)$) exhibiting, for sufficiently large $n$, a decision function whose risk was uniformly close to the envelope risk function of "simple" decision functions. The present paper considers the class of problems where the components involve decision between any two completely specified distributions, with the risk taken to be the weighted probability of wrong decision. For all sufficiently large $n$, decision functions are found whose risks are uniformly close to the envelope risk function of "invariant" decision functions.