On the estimation of a convex set. (English) Zbl 0557.62032

Let \(x_ 1,...,x_ n\) be independent and uniformly distributed on an unknown compact convex set D in \({\mathbb{R}}^ p\) (p known). The author considers the decision-theoretic estimation of the set D when the loss function is \(L(D,\hat D)=m(D\Delta \hat D)\) where \(\hat D\) is the estimate, m the Lebesgue measure and \(D\Delta\) \(\hat D\) is the symmetric difference between D and \(\hat D.\) Under suitable conditions he proves a complete class theorem for Bayes procedures. The cases \(p=1\) and \(p=2\) are discussed in detail, and the consistency of Bayes procedures is established.
Reviewer: R.A.Khan


62F15 Bayesian inference
62F99 Parametric inference
60D05 Geometric probability and stochastic geometry
62C07 Complete class results in statistical decision theory
62C10 Bayesian problems; characterization of Bayes procedures
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