Metric two-point risk bound ($R^{\ast }\ge \frac{r}{2}(1-\eta )(1-{\mathbb{E}}_0|Z-1|/\eta )$)

If two parameters are separated by at least $2r$ in a metric, any estimator gives a test by deciding whether its distance from the first parameter is below $r$. For $0<\eta <1$, the event $|Z-1|\le \eta$ and Markov inequality give the displayed minimax risk bound, where $Z$ is the product likelihood ratio. The unrestricted claim for every positive $\eta$ is false when both factors on the right are negative.