On the Elementary Theorems of Decision Theory.
FLORIDA STATE UNIV TALLAHASSEE DEPT OF STATISTICS
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This document considers a statistical decision problem in which nature has a finite number of states. The elementary theorems of decision theory, namely the Minimax theorem, the Complete class room theorem, and theorems on the structure of admissible rules, are proved in most texts under the assumptions that the risk set is closed from below and bounded from below. The condition that the risk set is bounded from below is sufficient for the existence of the lower boundary points however, that it is not necessary can be seen from simple examples. The purpose of this paper is to extend the elementary theorems of decision theory to include the case in which the risk set is not bounded from below and the set of lower boundary points is nonempty. The author also shows that if the risk set is bounded from above then it is necessary for the risk set to be bounded from below for the set of lower boundary points to be nonempty. Examples are presented to illustrate this theorems. Author
- Theoretical Mathematics