## The Annals of Statistics

#### Abstract

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.

#### Article information

Source
Ann. Statist., Volume 9, Number 4 (1981), 846-852.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176345524

Digital Object Identifier
doi:10.1214/aos/1176345524

Mathematical Reviews number (MathSciNet)
MR619287

Zentralblatt MATH identifier
0472.62013

JSTOR