Complete and Essentially Complete Classes

Abstract

We have previously observed that it is unwise to repeatedly use an inadmissible decision rule. (The possible exception is when an inadmissible rule is very simple and easy to use, and is only slightly inadmissible.) It is, therefore, of interest to find, for a given problem, the class of acceptable (usually admissible) decision rules. Such a class is often much easier to work with, say in finding a sequential Bayes, minimax or a r-minimax decision rule, than is the class of all decision rules. In this chapter, we discuss several of the most important situations in which simple reduced classes of decision rules have been obtained. Unfortunately, the subject tends to be quite difficult mathematically, and so we will be able to give only a cursory introduction to some of the more profound results. 8.1. Preliminaries We begin with definitions of the needed concepts. Definition 1. A class Cfi of decision rules is said to be essentially complete if, for any decision rule 8 not in Cfi, there is a decision rule 8' E Cfi which is R-better than or R-equivalent to 8. Definition 2. A class Cfi of decision rules is said to be complete if, for any decision rule 8 not in Cfi, there is a decision rule 8' E Cfi which is R-better than 8. Definition 3. A class Cfi of decision rules is said to be minimal complete if Cfi is complete and if no proper subset of Cfi is complete.