Suppose that $F_0$ is a population which is symmetric about zero, so that $F(\cdot) = F_0(\cdot - \theta)$ is symmetric about $\theta$. We consider the problem of estimating $F_0$ (shape parameter), both $\theta$ and $F_0$, and $F$ based on a random sample from $F$. First, some asymptotically minimax bounds are obtained. Then, some estimates are constructed which are asymptotically minimax-efficient (the risks of which achieve the minimax bounds uniformly). Furthermore, it is pointed out that one can estimate $F_0$, the shape of $F$, as well without knowing the location parameter $\theta$ as with knowing it. After a slight modification, Stone's (1975) estimator is proved to be asymptotically minimax-efficient in the Hellinger neighborhood.
"Estimation of a Symmetric Distribution." Ann. Statist. 13 (3) 1097 - 1113, September, 1985. https://doi.org/10.1214/aos/1176349658