## The Annals of Statistics

- Ann. Statist.
- Volume 12, Number 3 (1984), 1090-1099.

### On the Estimation of a Convex Set

#### Abstract

Given independent observations $x_1, \cdots, x_n$ drawn uniformly from an unknown compact convex set $D$ in $\mathbb{R}^p$ ($p$ known) it is desired to estimate $D$ from the observations. This problem was first considered, for $p = 2$, by Ripley and Rasson (1977). We consider a decision-theoretic approach where the loss function is $L(D, \hat{D}) = m(D \Delta \hat{D})$. We prove the completeness of the Bayes estimation rules. A form for the nonrandomized Bayes estimation rules is presented and applied, for an a priori law reflecting ignorance, to the cases $p = 1$ and where $D$ is a rectangle in the plane; some comparisons are made with other estimation methods suggested in the literature. Finally, the consistency of the estimation rules is studied.

#### Article information

**Source**

Ann. Statist., Volume 12, Number 3 (1984), 1090-1099.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176346725

**Digital Object Identifier**

doi:10.1214/aos/1176346725

**Mathematical Reviews number (MathSciNet)**

MR751296

**Zentralblatt MATH identifier**

0557.62032

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62F99: None of the above, but in this section

Secondary: 62F15: Bayesian inference 62C10: Bayesian problems; characterization of Bayes procedures 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

**Keywords**

Estimation convex set decision theory Bayesian complete convergence

#### Citation

Moore, Marc. On the Estimation of a Convex Set. Ann. Statist. 12 (1984), no. 3, 1090--1099. doi:10.1214/aos/1176346725. https://projecteuclid.org/euclid.aos/1176346725