High-dimensional Location Estimation via Norm Concentration for Subgamma Vectors

• URL: http://arxiv.org/abs/2302.02497v1
• Date: Sun, 5 Feb 2023 22:17:04 GMT
• Title: High-dimensional Location Estimation via Norm Concentration for Subgamma Vectors
• Authors: Shivam Gupta, Jasper C.H. Lee, Eric Price
• Abstract summary: In location estimation, we are given $n$ samples from a known distribution $f$ shifted by an unknown translation $lambda$. Asymptotically, the maximum likelihood estimate achieves the Cram'er-Rao bound of error $mathcal N(0, frac1nmathcal I)$. We build on the theory using emphsmoothed estimators to bound the error for finite $n$ in terms of $mathcal I_r$, the Fisher information of the $r$-smoothed
• Score: 15.802475232604667
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In location estimation, we are given $n$ samples from a known distribution $f$ shifted by an unknown translation $\lambda$, and want to estimate $\lambda$ as precisely as possible. Asymptotically, the maximum likelihood estimate achieves the Cram\'er-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 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.

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