In location estimation, we are given $n$ samples from a known distribution $f$ shifted by an unknown translation $\lambda$, and want to estimate $\lambda$ as precisely as possible. Asymptotically, the maximum likelihood estimate achieves the Cram\'er-Rao bound of error $\mathcal N(0, \frac{1}{n\mathcal I})$, where $\mathcal I$ is the Fisher information of $f$. However, the $n$ required for convergence depends on $f$, and may be arbitrarily large. We build on the theory using \emph{smoothed} estimators to bound the error for finite $n$ in terms of $\mathcal I_r$, the Fisher information of the $r$-smoothed distribution. As $n \to \infty$, $r \to 0$ at an explicit rate and this converges to the Cram\'er-Rao bound. We (1) improve the prior work for 1-dimensional $f$ to converge for constant failure probability in addition to high probability, and (2) extend the theory to high-dimensional distributions. In the process, we prove a new bound on the norm of a high-dimensional random variable whose 1-dimensional projections are subgamma, which may be of independent interest.

翻译：在位置估计问题中，给定来自已知分布$f$（经未知平移量$\lambda$偏移）的$n$个样本，目标是尽可能精确地估计$\lambda$。渐近地，最大似然估计达到Cramér-Rao误差界$\mathcal N(0, \frac{1}{n\mathcal I})$，其中$\mathcal I$为$f$的Fisher信息量。然而，收敛所需的$n$取决于$f$，且可能任意大。我们基于\emph{平滑}估计量理论，利用$r$-平滑分布的Fisher信息量$\mathcal I_r$来界定有限$n$情况下的误差。当$n \to \infty$时，$r$以显式速率趋于$0$，该界收敛至Cramér-Rao界。我们(1)将一维$f$的先前工作改进为在常数失败概率和高概率下均收敛，且(2)将该理论推广至高维分布。在此过程中，我们证明了一维投影为子伽马的高维随机变量范数的新上界，该结果可能具有独立研究价值。