3.2 Classical Decision Theory

3.2.2 Admissibility

Definition 3.24

• $\mathcal{C}$,
• $D$
  • decision set
• $\delta: \mathcal{X} \rightarrow D$,
  • decision function
• $\delta^{\prime}: \mathcal{X} \rightarrow D$,
  • decision function

$\delta_{1}$ is said to dominate $\delta$ if

\[\begin{eqnarray} \forall \theta, \ R(\theta, \delta_{1}) & \le & R(\theta, \delta) \nonumber \\ \exists \theta \in \Omega, \text{ s.t. } R(\theta, \delta_{1}) & < & R(\theta, \delta) \nonumber . \end{eqnarray}\]

$\delta$ is inadmissible in $\mathcal{C}$ if there exists another desicision rule $\delta_{1}$ such that $\delta_{1}$ dominates $\delta$.

\[\exists \delta_{1} \in \mathcal{C} \text{ s.t. } \delta_{1} \text{ dominates } \delta.\]

$\delta$ is adminissible in $\mathcal{C}$ if there is no decision rule which dominates $\delta$.

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3.2.5 Complete Classes

Definition 3.83 complete class

• $\mathcal{C}$
  • set of decision rules

$\mathcal{C}$ is complete if

\[\forall \delta \notin \mathcal{C}, \ \exists \delta_{0} \in \mathcal{C}, \ \text{ s.t. } \ \delta_{0} \text{ dominates } \delta .\]

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