Finite-Sample Maximum Likelihood Estimation of Location
release_5uunoegv6nakdclc5l2sbcngoy
by
Shivam Gupta, Jasper C.H. Lee, Eric Price, Paul Valiant
2022
Abstract
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 →∞: it is
asymptotically normal with variance matching the Cramér-Rao lower bound of
1/nℐ, where 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.
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