Probability Theory and Related Fields

, Volume 175, Issue 3–4, pp 957–973 | Cite as

Near-optimal mean estimators with respect to general norms

  • Gábor LugosiEmail author
  • Shahar Mendelson


We study the problem of estimating the mean of a random vector in \(\mathbb {R}^d\) based on an i.i.d. sample, when the accuracy of the estimator is measured by a general norm on \(\mathbb {R}^d\). We construct an estimator (that depends on the norm) that achieves an essentially optimal accuracy/confidence tradeoff under the only assumption that the random vector has a well-defined covariance matrix. At the heart of the argument is the construction of a uniform median-of-means estimator in a class of real valued functions.

Mathematics Subject Classification

62H12 62G05 



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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Economics and BusinessPompeu Fabra UniversityBarcelonaSpain
  2. 2.ICREABarcelonaSpain
  3. 3.Barcelona Graduate School of EconomicsBarcelonaSpain
  4. 4.LPSMSorbonne UniversitéParisFrance
  5. 5.Mathematical Sciences InstituteThe Australian National UniversityCanberraAustralia

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