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September, 1984 On the Estimation of a Convex Set
Marc Moore
Ann. Statist. 12(3): 1090-1099 (September, 1984). DOI: 10.1214/aos/1176346725


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.


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Marc Moore. "On the Estimation of a Convex Set." Ann. Statist. 12 (3) 1090 - 1099, September, 1984.


Published: September, 1984
First available in Project Euclid: 12 April 2007

zbMATH: 0557.62032
MathSciNet: MR751296
Digital Object Identifier: 10.1214/aos/1176346725

Primary: 62F99
Secondary: 60D05, 62C10, 62F15

Rights: Copyright © 1984 Institute of Mathematical Statistics


Vol.12 • No. 3 • September, 1984
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