Admissibility in Finite Problems

Meeden, Glen ; Ghosh, Malay

Ann. Statist., Tome 9 (1981) no. 1, p. 846-852 / Harvested from Project Euclid

Résumé

Let $X$ be a random variable which takes on only finitely many values $x \in \chi$ with a finite family of possible distributions indexed by some parameter $\theta \in \Theta$. Let $\Pi = \{\pi_x(\cdot):x \in \chi\}$ be a family of possible distributions (termed "inverse probability distributions") on $\Theta$ depending on $x \in \chi$. A theorem is given to characterize the admissibility of a decision rule $\delta$ which minimizes the expected loss with respect to the distribution $\pi_x(\cdot)$ for each $x \in \chi$. The theorem is partially extended to the case when the sample space and the parameter space are not necessarily finite. Finally a notion of "admissible consistency" is introduced and a necessary and sufficient condition for admissible consistency is provided when the parameter space is finite, while the sample space is countable.

Détail

Publié le : 1981-07-14
Classification:  Inverse probability distributions,  admissibility,  Bayes rules,  singular priors,  admissible consistency,  expectation consistency,  discrete uniform,  squared error loss,  62C15,  62F10