 Solution

Apply the preceding argument to the two product observation laws and the real-valued estimator $T={\hat{\psi }}_n$. Alternatively, directly useThe average of the two mean squared errors is at least $(\theta -\tau {)}^2/4$ times the overlap of the two joint probability density functions. Their overlap equalsTaking the infimum over estimators therefore yieldsHere $R_n^{\ast }$ is the minimax risk for this functional. The factor $\frac{1}{2}\Vert P_f^{(n)}-P_g^{(n)}{\Vert }_1$ is the total variation distance, so the same metric squared-loss two-point bound applies to a functional even when distinct density parameters have the same functional value.