Finite-Sample Maximum Likelihood Estimation of Location. (arXiv:2206.02348v1 [math.ST])

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 …

arxiv likelihood location math maximum likelihood estimation