## The Annals of Statistics

- Ann. Statist.
- Volume 6, Number 4 (1978), 820-827.

### Complete Class Theorems Derived from Conditional Complete Class Theorems

#### Abstract

Let $(\mathscr{X}, \mathscr{B}_1, \mu)$ and $(\mathscr{Y}, \mathscr{B}_2, \nu)$ be $\sigma$-finite measure spaces and suppose $\Theta$ is a separable metric space. Let $f(x \mid y, \theta)$ be a family of conditional densities on $(\mathscr{X}, \mathscr{B}, \mu).$ Consider an action space $A$ which is a compact metric space with $\mathscr{B}_A$ the Borel $\sigma$-algebra and a loss function $W(\theta, a)$ such that $W(\theta, \bullet)$ is continuous. For any decision rule $\delta: \mathscr{B}_A \times \mathscr{X} \rightarrow \lbrack 0, 1\rbrack,$ assume the risk function $R(\delta, \bullet)$ is continuous on $\Theta.$ Suppose that a set of decision rules $\mathscr{M}_0$ is an essentially complete class for each $y \in \mathscr{Y}$ for the conditional decision problem. Let $\mathscr{M}^\ast$ be the set of decision rules $\eta: \mathscr{B}_A \times (\mathscr{X} \times \mathscr{Y}) \rightarrow \lbrack 0, 1\rbrack$ such that $\eta(\bullet \mid \bullet, y) \in \mathscr{M}_0 \mathrm{a.e.} \lbrack \nu\rbrack.$ Then $\mathscr{M}^\ast$ is an essentially complete class no matter what the family of marginal densities on the space $(\mathscr{Y}, \mathscr{B}_2, \nu).$

#### Article information

**Source**

Ann. Statist., Volume 6, Number 4 (1978), 820-827.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176344255

**Digital Object Identifier**

doi:10.1214/aos/1176344255

**Mathematical Reviews number (MathSciNet)**

MR494595

**Zentralblatt MATH identifier**

0378.62002

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62C07: Complete class results

**Keywords**

Decision theory complete class conditional complete class

#### Citation

Eaton, Morris L. Complete Class Theorems Derived from Conditional Complete Class Theorems. Ann. Statist. 6 (1978), no. 4, 820--827. doi:10.1214/aos/1176344255. https://projecteuclid.org/euclid.aos/1176344255