A constrained risk inequality for general losses

04/22/2018

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by   John C. Duchi, et al.

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We provide a general constrained risk inequality that applies to arbitrary non-decreasing losses, extending a result of Brown and Low [Ann. Stat. 1996]. Given two distributions P_0 and P_1, we find a lower bound for the risk of estimating a parameter θ(P_1) under P_1 given an upper bound on the risk of estimating the parameter θ(P_0) under P_0. As our inequality applies to general losses, it allows further insights on super-efficiency and adaptive estimators.