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

This defines a partial order on decision rules; the maximal elements with respect to this order are called admissible decision rules.¹