Sequential Compound Estimators

Abstract

This paper deals with the sequential compound decision problem, when the component problem is an estimation problem. The aim is to find sequential compound rules which have Property (A) or (B), as described in Section 2. That is, one seeks rules the compound risk or loss of which do not exceed the value of the Bayes envelope functional at the "empirical distribution" of the parameters, by more than $\epsilon$, for $n$ sufficiently large. In Section 4 some general results for the estimation problem are obtained, which are applied in Sections 5 and 6 to several discrete and continuous distributions to obtain rules with Property (A) or (B). In Section 3 some results are obtained which hold for the sequential compound decision problem with any general component problem. These are applied for the estimation problem in the later sections.