## 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

### MSC:

 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

### Keywords:

estimation of convex sets; loss function; consistency
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