Open Access
April 2020 Mean estimation with sub-Gaussian rates in polynomial time
Samuel B. Hopkins
Ann. Statist. 48(2): 1193-1213 (April 2020). DOI: 10.1214/19-AOS1843


We study polynomial time algorithms for estimating the mean of a heavy-tailed multivariate random vector. We assume only that the random vector $X$ has finite mean and covariance. In this setting, the radius of confidence intervals achieved by the empirical mean are large compared to the case that $X$ is Gaussian or sub-Gaussian.

We offer the first polynomial time algorithm to estimate the mean with sub-Gaussian-size confidence intervals under such mild assumptions. Our algorithm is based on a new semidefinite programming relaxation of a high-dimensional median. Previous estimators which assumed only existence of finitely many moments of $X$ either sacrifice sub-Gaussian performance or are only known to be computable via brute-force search procedures requiring time exponential in the dimension.


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Samuel B. Hopkins. "Mean estimation with sub-Gaussian rates in polynomial time." Ann. Statist. 48 (2) 1193 - 1213, April 2020.


Received: 1 October 2018; Revised: 1 February 2019; Published: April 2020
First available in Project Euclid: 26 May 2020

zbMATH: 07241586
MathSciNet: MR4102693
Digital Object Identifier: 10.1214/19-AOS1843

Primary: 62H12
Secondary: 68W

Keywords: confidence intervals , heavy tails , multivariate estimation , semidefinite programming , sub-Gaussian rates , sum of squares method

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 2 • April 2020
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