Exercise 15.8: Sharper Bounds for Gaussian Location Families
Let $\mathcal{P} = \lbrace \gauss (\theta, \sigma^2) \colon \theta \in \R \rbrace$, and observe that for any $q \geq 1$, by the 2-point Le Cam \begin{align} \inf_{\hat{\theta}} \sup_{\theta} \E | \hat{\theta} - \theta |^q \geq \tfrac{\delta^q}{2} (1 - \TV (P_0, P_1)) \, , \end{align} for any $P_0, P_1 \in \mathcal{P}$ with $|\theta_0 - \theta_1 | \geq 2 \delta$. Then by the Pinsker-Csiszar-Kullback inequality \begin{align} \TV(P_0, P_1) \leq \sqrt{\tfrac{1}{2} \KL (P_0 \, \| \, P_1)} = \tfrac{1}{2 \sigma} | \theta_0 - \theta_1 | \, , \end{align} where we substituted the known expression for KL divergence between two Gaussian distributions.
Applying both inequalities consequtively, and choosing $\delta = \tfrac{\sigma}{2 \sqrt{n}}$ together with $| \theta_0 - \theta_1 | = 2 \delta$ yields the respective results for $q = 1$ and $q = 2$.