Finite-Sample Maximum Likelihood Estimation of Location

06/06/2022

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by   Shivam Gupta, et al.

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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.