 Le Cam lower bound under absolute-error loss

For two observation laws $P_0,P_1$ at real parameters separated by $\Delta$, every estimator $T$ satisfies ${\max }_j{\mathbb{E}}_j|T-{\theta }_j|\ge \Delta (1-\mathrm{TV}(P_0,P_1))/2$. With common densities, the sum of the two risks is at least $\int \min (p_0,p_1)(|T-{\theta }_0|+|T-{\theta }_1|)$, at least $\Delta \int \min (p_0,p_1)=\Delta (1-\mathrm{TV})$ by the triangle inequality. Divide by two. The same proof works for any metric loss.