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Poster
High-dimensional Location Estimation via Norm Concentration for Subgamma Vectors
Shivam Gupta · Jasper Lee · Eric Price

Wed Jul 26 02:00 PM -- 03:30 PM (PDT) @ Exhibit Hall 1 #624
in Poster Session 3 »
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 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 *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.
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Author Information

Shivam Gupta (University of Texas at Austin)
Jasper Lee (University of Wisconsin-Madison)
Eric Price (UT-Austin)

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