The Relation between Statistical Decision Theory and Approximation Theory.
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WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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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.
- Statistics and Probability