Finite-Sample Maximum Likelihood Estimation of Location

Abstract

We consider 1-dimensional location estimation, where we estimate a parameter $\hat{I}»$ from $n$ samples $\hat{I}»+\hat{I}{\cdot }_i$, with each $\hat{I}{\cdot }_i$ 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 $n$.

Corrected an inaccuracy in the description of the experimental setup. Also updated funding acknowledgements