Characterize adaptive distribution classes where two-point rates are attainable

Let $\mathcal F$ be a nonempty class of probability distributions on $\mathbb R$ such that every $P\in\mathcal F$ has finite mean $\mu(P):=\int x\,dP(x)$, and $\mathcal F$ is translation-invariant in the sense that for every $P\in\mathcal F$ and every $t\in\mathbb R$, the translated law $P_t$ of $X+t$ (for $X\sim P$) also belongs to $\mathcal F$. For $P,Q$ on $\mathbb R$, define squared Hellinger distance by

where $\lambda$ is any common dominating measure (the value is independent of $\lambda$). For $m\in\mathbb N$, let $P^{\otimes m}$ denote the $m$-fold product law, and define the local $m$-sample Hellinger modulus for mean estimation at $P$ by

(Any fixed constant in $(0,1)$ in place of $\tfrac14$ is equivalent up to absolute-constant rescaling of sample size.)

Given i.i.d. data $X_1,\dots,X_n\sim P$ with unknown $P\in\mathcal F$, an estimator is any measurable map $\hat\mu_n=\hat\mu_n(X_1,\dots,X_n)\in\mathbb R$.