Dominating decision rule

In decision theory, a decision rule is said to dominate another if the performance of the former is sometimes better, and never worse, than that of the latter.

Formally, let $\delta _{1}$ and $\delta _{2}$ be two decision rules, and let $R(\theta ,\delta )$ be the risk of rule $\delta$ for parameter $\theta$. The decision rule $\delta _{1}$ is said to dominate the rule $\delta _{2}$ if $R(\theta ,\delta _{1})\leq R(\theta ,\delta _{2})$ for all $\theta$, and the inequality is strict for some $\theta$.

This defines a partial order on decision rules; the maximal elements with respect to this order are called admissible decision rules.