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# Statistical minimax theorems via nonstandard analysis

Dec 2022

For statistical decision problems with finite parameter space, it iswell-known that the upper value (minimax value) agrees with the lower value(maximin value). Only under a generalized notion of prior does such anequivalence carry over to the case infinite parameter spaces, provided naturecan play a prior distribution and the statistician can play a randomizedstrategy. Various such extensions of this classical result have beenestablished, but they are subject to technical conditions such as compactnessof the parameter space or continuity of the risk functions. Using nonstandardanalysis, we prove a minimax theorem for arbitrary statistical decisionproblems. Informally, we show that for every statistical decision problem, thestandard upper value equals the lower value when the \$\sup\$ is taken over thecollection of all internal priors, which may assign infinitesimal probabilityto (internal) events. Applying our nonstandard minimax theorem, we deriveseveral standard minimax theorems: a minimax theorem on compact parameter spacewith continuous risk functions, a finitely additive minimax theorem withbounded risk functions and a minimax theorem on totally bounded metricparameter spaces with Lipschitz risk functions.

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