Finite-Sample Maximum Likelihood Estimation of Location

TL;DR

This paper develops a finite-sample theory for maximum likelihood estimation of a location parameter, showing how to approximate the asymptotic variance using a smoothed version of the distribution, applicable for finite samples.

Contribution

It introduces a finite-sample analysis of MLE for location estimation using a smoothed Fisher information, extending classical asymptotic results to finite sample sizes.

Findings

Finite-sample MLE variance approximated by smoothed Fisher information.

The theory applies to arbitrary distributions and finite sample sizes.

Asymptotic normality is extended to finite samples with smoothing.

Abstract

We consider 1-dimensional location estimation, where we estimate a parameter $\lambda$ from $n$ samples $\lambda + \eta_i$, with each $\eta_i$ 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 Cram\'er-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 $n$.