Abstract:
We provide a general constrained risk inequality that applies to arbitrary
non-decreasing losses, extending a result of Brown and Low
[\emph{Ann.~Stat.~1996}]. Given two distributions $P_0$ and $P_1$, we find
a lower bound for the risk of estimating a parameter $\theta(P_1)$ under
$P_1$ given an upper bound on the risk of estimating the parameter
$\theta(P_0)$ under $P_0$. The inequality is a useful pedagogical tool, as
its proof relies only on the Cauchy-Schwartz inequality, it applies to
general losses, and it transparently gives risk lower bounds on
super-efficient and adaptive estimators.