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 $$R(\theta,\delta_1)\le R(\theta,\delta_2)$$ 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.