Statistical minimax theorems via nonstandard analysis
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For statistical decision problems with finite parameter space, it is well-known that the upper value (minimax value) agrees with the lower value (maximin value). Only under a generalized notion of prior does such an equivalence carry over to the case infinite parameter spaces, provided nature can play a prior distribution and the statistician can play a randomized strategy. Various such extensions of this classical result have been established, but they are subject to technical conditions such as compactness of the parameter space or continuity of the risk functions. Using nonstandard analysis, we prove a minimax theorem for arbitrary statistical decision problems. Informally, we show that for every statistical decision problem, the standard upper value equals the lower value when the sup is taken over the collection of all internal priors, which may assign infinitesimal probability to (internal) events. Applying our nonstandard minimax theorem, we derive several standard minimax theorems: a minimax theorem on compact parameter space with continuous risk functions, a finitely additive minimax theorem with bounded risk functions and a minimax theorem on totally bounded metric parameter spaces with Lipschitz risk functions.
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