High-dimensional Location Estimation via Norm Concentration for Subgamma Vectors

Abstract

In location estimation, we are given n samples from a known distribution f shifted by an unknown translation , and want to estimate as precisely as possible. Asymptotically, the maximum likelihood estimate achieves the Cram\'er-Rao bound of error N(0, 1n I), where I is the Fisher information of f. However, the n required for convergence depends on f, and may be arbitrarily large. We build on the theory using smoothed estimators to bound the error for finite n in terms of I_r, the Fisher information of the r-smoothed distribution. As n , r 0 at an explicit rate and this converges to the Cram\'er-Rao bound. We (1) improve the prior work for 1-dimensional f to converge for constant failure probability in addition to high probability, and (2) extend the theory to high-dimensional distributions. In the process, we prove a new bound on the norm of a high-dimensional random variable whose 1-dimensional projections are subgamma, which may be of independent interest.

Tasks

Reproductions