This paper is devoted to the statistical properties of the geometric median, a robust measure of centrality for multivariate data, as well as its applications to the problem of mean estimation via the median of means principle.
Our main theoretical results include (a) the upper bound for the distance between the mean and the median for general absolutely continuous distributions in $\mathbb R^d$, and examples of specific classes of distributions for which these bounds do not depend on the ambient dimension $d$; (b) exponential deviation inequalities for the distance between the sample and the population versions of the geometric median, which again depend only on the trace-type quantities and not on the ambient dimension. As a corollary, we deduce the improved bounds for the multivariate median of means estimator that hold for large classes of heavy-tailed distributions.
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U-statistics of growing order and sub-Gaussian mean estimators with sharp constants
This paper addresses the following question: given a sample of i.i.d. random variables with finite variance, can one construct an estimator of the unknown mean that performs nearly as well as if the data were normally distributed? One of the most popular examples achieving this goal is the median of means estimator. However, it is inefficient in a sense that the constants in the resulting bounds are suboptimal. We show that a permutation-invariant modification of the median of means estimator admits deviation guarantees that are sharp up to $1+o(1)$ factor if the underlying distribution possesses more than $\frac{3+\sqrt{5}}{2}\approx 2.62$ moments and is absolutely continuous with respect to the Lebesgue measure. This result yields potential improvements for a variety of algorithms that rely on the median of means estimator as a building block.
At the core of our argument is are the new deviation inequalities for the U-statistics of order that is allowed to grow with the sample size, a result that could be of independent interest.
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- NSF-PAR ID:
- 10423177
- Date Published:
- Journal Name:
- Mathematical statistics and learning
- ISSN:
- 2520-2324
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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The goal of this note is to present a modification of the popular median of means estimator that achieves sub-Gaussian deviation bounds with nearly optimal constants under minimal assumptions on the underlying distribution. We build on the recent work on the topic and prove that desired guarantees can be attained under weaker requirements.more » « less
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- The DOI is not currently available.
