Sharp boundary for attainability in finite-sample location models

Let $P_0$ be a Borel probability distribution on $\mathbb{R}$ with finite first moment and centered so that $\int x\,dP_0(x)=0$. For each shift parameter $\theta\in\mathbb{R}$, define the location family member $P_\theta$ by $P_\theta(A)=P_0(A-\theta)$ for all Borel sets $A\subseteq\mathbb{R}$. Given $n$ i.i.d. observations $X_1,\dots,X_n\sim P_{\theta^\star}$ with unknown $\theta^\star$, an estimator is any measurable map $\hat\theta_n:\mathbb{R}^n\to\mathbb{R}$.

Define worst-case absolute-error risk

and minimax risk $R_n^\star(P_0)=\inf_{\hat\theta_n}R_n(\hat\theta_n;P_0)$.

For probability measures $P,Q$ on $\mathbb{R}$, define squared Hellinger distance

where $\mu$ is any dominating measure (this value is independent of $\mu$). Define the location-model Hellinger modulus for real effective sample size $t\ge 1$ by

(For integer $t=m$, this matches the usual sample-size-$m$ two-point-testing benchmark up to universal constants via Le Cam lower bounds.)

Say the two-point rate is near-attainable for this fixed base distribution $P_0$ if there exist constants $C>0$, $a,b\ge 0$, and estimators $\{\hat\theta_n\}_{n\ge 2}$ such that for all $n\ge 2$,

Equivalently: for this $P_0$, estimation error matches the Hellinger-modulus lower bound up to polylogarithmic factors and a $\tilde O(n)$ effective sample-size loss.