The Relation between Statistical Decision Theory and Approximation Theory.

摘要

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 phi(theta) is the risk function of the statistical decision procedure phi. We interpret S as a set in the normed linear space L. Let v=v(theta) satisfy v(theta) < 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.