On the estimation of a convex set
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.
scientific article; zbMATH DE number 3888712