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

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