The Annals of Statistics

Online estimation of the geometric median in Hilbert spaces: Nonasymptotic confidence balls

Hervé Cardot, Peggy Cénac, and Antoine Godichon-Baggioni

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Abstract

Estimation procedures based on recursive algorithms are interesting and powerful techniques that are able to deal rapidly with very large samples of high dimensional data. The collected data may be contaminated by noise so that robust location indicators, such as the geometric median, may be preferred to the mean. In this context, an estimator of the geometric median based on a fast and efficient averaged nonlinear stochastic gradient algorithm has been developed by [Bernoulli 19 (2013) 18–43]. This work aims at studying more precisely the nonasymptotic behavior of this nonlinear algorithm by giving nonasymptotic confidence balls in general separable Hilbert spaces. This new result is based on the derivation of improved $L^{2}$ rates of convergence as well as an exponential inequality for the nearly martingale terms of the recursive nonlinear Robbins–Monro algorithm.

Article information

Source
Ann. Statist., Volume 45, Number 2 (2017), 591-614.

Dates
Received: January 2015
Revised: February 2016
First available in Project Euclid: 16 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.aos/1494921951

Digital Object Identifier
doi:10.1214/16-AOS1460

Mathematical Reviews number (MathSciNet)
MR3650394

Zentralblatt MATH identifier
1371.62027

Subjects
Primary: 62G05: Estimation
Secondary: 62L20: Stochastic approximation

Keywords
Functional data analysis martingales in Hilbert spaces recursive estimation robust statistics spatial median stochastic gradient algorithms

Citation

Cardot, Hervé; Cénac, Peggy; Godichon-Baggioni, Antoine. Online estimation of the geometric median in Hilbert spaces: Nonasymptotic confidence balls. Ann. Statist. 45 (2017), no. 2, 591--614. doi:10.1214/16-AOS1460. https://projecteuclid.org/euclid.aos/1494921951


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Supplemental materials

  • Supplement to “Online estimation of the geometric median in Hilbert spaces: Nonasymptotic confidence balls”. We provide the proofs of some technical ancillary lemmas and propositions.