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We provide a general constrained risk inequality that applies to arbitrarynon-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 ofestimating a parameter $\theta(P_1)$ under $P_1$ given an upper bound on therisk of estimating the parameter $\theta(P_0)$ under $P_0$. The inequality is auseful pedagogical tool, as its proof relies only on the Cauchy-Schwartzinequality, it applies to general losses, and it transparently gives risk lowerbounds on super-efficient and adaptive estimators.

A constrained risk inequality for general losses