Statistical decisions under ambiguity

Statistical decisions under ambiguity

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Article ID: iaor20111430
Volume: 70
Issue: 2
Start Page Number: 129
End Page Number: 148
Publication Date: Feb 2011
Journal: Theory and Decision
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Abstract:

This article provides unified axiomatic foundations for the most common optimality criteria in statistical decision theory. It considers a decision maker who faces a number of possible models of the world (possibly corresponding to true parameter values). Every model generates objective probabilities, and von Neumann–Morgenstern expected utility applies where these obtain, but no probabilities of models are given. This is the classic problem captured by Wald's (1950) device of risk functions. In an Anscombe–Aumann environment, I characterize Bayesianism (as a backdrop), the statistical minimax principle, the Hurwicz criterion, minimax regret, and the ‘Pareto’ preference ordering that rationalizes admissibility. Two interesting findings are that c‐independence is not crucial in characterizing the minimax principle and that the axiom which picks minimax regret over maximin utility is von Neumann–Morgenstern independence.

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