1.8 Complete Classes

Definition admissible

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

$\delta$ is said to be inadmissible if there exists $\delta^{\prime}$ such that

\[\begin{eqnarray} \forall \theta \in \Theta, & & R(\theta, \delta^{\prime}) \le R(\theta, \delta) \nonumber \\ \exists \theta \in \Theta & \text{ s.t. } & R(\theta, \delta^{\prime}) < R(\theta, \delta) . \label{chap01_08_01_15} \end{eqnarray}\]

$\delta$ is said to be admissible if $\delta$ is not inadmissible.

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Definition Complete

• $\mathcal{D}$,
  • decision functions
• $\mathcal{C} \subseteq \mathcal{D}$,
  • a set of decision function.

$\mathcal{C}$ is said to be complete if for any $\delta \in \mathcal{C}$ there exists $\delta^{\prime} \in \mathcal{C}$ such that \(\eqref{chap01_08_01_15}\) is satisfied.

A complete class $\mathcal{C}$ is said to be minimal if $\mathcal{C}$ is the smallest subset of $\mathcal{D}$ which is complete.

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Definition essentially complete

• $\mathcal{D}$,
  • decision functions
• $\mathcal{C} \subseteq \mathcal{D}$,
  • a set of decision function.

$\mathcal{C}$ is said to be essentially complete if

\[\forall \delta \in \mathcal{D}, \ \exists \delta^{\prime} \in \mathcal{C} \text{ s.t. } \forall \theta \in \Theta, \ R(\theta, \delta^{\prime}) \le R(\theta, \delta) .\]

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