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April 2017 Online estimation of the geometric median in Hilbert spaces: Nonasymptotic confidence balls
Hervé Cardot, Peggy Cénac, Antoine Godichon-Baggioni
Ann. Statist. 45(2): 591-614 (April 2017). DOI: 10.1214/16-AOS1460

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.

Citation

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Hervé Cardot. Peggy Cénac. Antoine Godichon-Baggioni. "Online estimation of the geometric median in Hilbert spaces: Nonasymptotic confidence balls." Ann. Statist. 45 (2) 591 - 614, April 2017. https://doi.org/10.1214/16-AOS1460

Information

Received: 1 January 2015; Revised: 1 February 2016; Published: April 2017
First available in Project Euclid: 16 May 2017

zbMATH: 1371.62027
MathSciNet: MR3650394
Digital Object Identifier: 10.1214/16-AOS1460

Subjects:
Primary: 62G05
Secondary: 62L20

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

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 2 • April 2017
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