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Data-based decisions under imprecise probability and least favorable models. (English) Zbl 1185.62021

Summary: Data-based decision theory under imprecise probability has to deal with optimization problems where direct solutions are often computationally intractable. Using the \(\Gamma \)-minimax optimality criterion, the computational effort may significantly be reduced in the presence of a least favorable model. A. Buja [Simultaneously least favorable experiments. I. Upper standard functionals and sufficiency. Wahrscheinlichkeitstheor. Verw. Gebiete 65, 367–384 (1984; Zbl 0506.62002)] derived a necessary and sufficient condition for the existence of a least favorable model in a special case. The present article proves that essentially the same result is valid in the case of general coherent upper previsions. This is done mainly by topological arguments in combination with some of L. Le Cam’s decision theoretic concepts [see “Asymptotic methods in statistical decision theory.” NY: Springer (1986; Zbl 0605.62002)]. It is shown how least favorable models could be used to deal with situations where the distribution of the data as well as the prior is allowed to be imprecise.

MSC:

62C99 Statistical decision theory
62C20 Minimax procedures in statistical decision theory
62C10 Bayesian problems; characterization of Bayes procedures
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