论文标题
有限样本的最大似然位置估计
Finite-Sample Maximum Likelihood Estimation of Location
论文作者
论文摘要
我们考虑一维位置估计，其中我们从$ n $ samples $λ+η_i$估算一个参数$λ$，每个$η_i$ drawn i.i.d.从已知的分销$ f $。对于固定的$ f $，最大易变估计值（MLE）在限制中以$ n \ to \ infty $的限制是最佳的：它在零散正常上与cramér-rao匹配的$ \ frac {1} {n \ mathcal {i}} $ is $ is $ noff $ \ noffer $ \ \ \ \ fcramér-rao的下限。但是，这种界限不适合有限$ n $，或者当$ f $随$ n $而变化时。我们以任意$ f $和$ n $的方式显示，人们可以根据$ f $的平滑版本的渔民信息来恢复类似的理论，在这种情况下，平滑半径损失了$ n $。
We consider 1-dimensional location estimation, where we estimate a parameter $λ$ from $n$ samples $λ+ η_i$, with each $η_i$ drawn i.i.d. from a known distribution $f$. For fixed $f$ the maximum-likelihood estimate (MLE) is well-known to be optimal in the limit as $n \to \infty$: it is asymptotically normal with variance matching the Cramér-Rao lower bound of $\frac{1}{n\mathcal{I}}$, where $\mathcal{I}$ is the Fisher information of $f$. However, this bound does not hold for finite $n$, or when $f$ varies with $n$. We show for arbitrary $f$ and $n$ that one can recover a similar theory based on the Fisher information of a smoothed version of $f$, where the smoothing radius decays with $n$.
