Decision Theory

8Admissibility

In this chapter we will explore further the concept of admissibility. Suppose, following Wald, we agree to look at long-run average loss R as the criterion of interest for choosing decision rules. R depends on θ, but a basic requirement is that one should not prefer a rule that does worse than another no matter what the true θ is. This is a very weak requirement, and a key rationality principle for frequentist decision theory. There is a similarity between admissibility and the strict coherence condition presented in Section 2.1.2. In de Finetti’s terminology, a decision maker trading a decision rule for another that has higher risk everywhere could be described as a sure loser—except for the fact that the risk difference could be zero in some “lucky” cases.

Admissible rules are those that cannot be dominated; that is, beaten at the risk game no matter what the truth is. Admissibility is a far more basic rationality principle than minimax in the sense that it does not require adhering to the “ultra-pessimistic” perspective on θ. The group of people willing to take admissibility seriously is in fact much larger than those equating minimax to frequentist rationality. To many, therefore, characterization of sets of admissible decision rules has been a key component of the contribution of decision theory to statistics.

It turns out that just by requiring admissibility one is drawn again towards an expected utility perspective. We will discover that to generate an admissible ...