## The Annals of Statistics

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

### Admissibility in Finite Problems

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 62C15: Admissibility

Secondary: 62F10: Point estimation

**Keywords**

Inverse probability distributions admissibility Bayes rules singular priors admissible consistency expectation consistency discrete uniform squared error loss

#### Citation

Meeden, Glen; Ghosh, Malay. Admissibility in Finite Problems. Ann. Statist. 9 (1981), no. 4, 846--852. doi:10.1214/aos/1176345524. https://projecteuclid.org/euclid.aos/1176345524