High-dimensional Location Estimation via Norm Concentration for Subgamma Vectors
This work addresses the challenge of precise location estimation in statistics, particularly for high-dimensional data, with incremental improvements over existing methods.
The paper tackles the problem of location estimation for finite sample sizes by developing smoothed estimators that bound error in terms of Fisher information, improving prior work to achieve constant failure probability in one dimension and extending it to high-dimensional distributions.
In location estimation, we are given $n$ samples from a known distribution $f$ shifted by an unknown translation $λ$, and want to estimate $λ$ as precisely as possible. Asymptotically, the maximum likelihood estimate achieves the Cramér-Rao bound of error $\mathcal N(0, \frac{1}{n\mathcal I})$, where $\mathcal I$ 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 \emph{smoothed} estimators to bound the error for finite $n$ in terms of $\mathcal I_r$, the Fisher information of the $r$-smoothed distribution. As $n \to \infty$, $r \to 0$ at an explicit rate and this converges to the Cramér-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.