WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
The approximation theory model describes a class of optimality principles in statistical decision theory as follows. Let S be the risk set of a statistical decision problem, that is, S R sub phi theta, theta an element of Theta, phi an element of Phi where Phi is the collection of randomized decision procedures, Theta is the parameter space and R sub phitheta is the risk function of the statistical decision procedure phi. We interpret S as a set in the normed linear space L. Let vvtheta satisfy vtheta or R sub phi for all phi an element of Phi and all theta an element of Theta. Then s sub 0 an element of S is said to be v,L optimal if abs. val. s sub 0-v or abs. val. s-v for all s an element of S. It is easily seen that many well-known optimality principles of statistics are of this type, such as Bayes rules and minimax rules. In this paper, characterization theorems for this class of optimality principles are given.
Technical summary rept.,
Prepared in cooperation with Munich Univ. (West Germany). Mathematisches Inst. der Technischen.