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

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 prefered 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.
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Annals of Statistics, Institute of Mathematical Statistics, 2017, 45 (2), pp.591 - 614. 〈10.1214/16-AOS1460〉
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Contributeur : Imb - Université de Bourgogne <>
Soumis le : mardi 4 juillet 2017 - 11:45:07
Dernière modification le : jeudi 11 janvier 2018 - 06:12:20

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Hervé Cardot, Peggy Cénac, Antoine Godichon-Baggioni. Online estimation of the geometric median in Hilbert spaces: Nonasymptotic confidence balls. Annals of Statistics, Institute of Mathematical Statistics, 2017, 45 (2), pp.591 - 614. 〈10.1214/16-AOS1460〉. 〈hal-01555593〉

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