Estimating the geometric median in Hilbert spaces with stochastic gradient algorithms: L p and almost sure rates of convergence

The geometric median, also called $L^1$-median, is often used in robust statistics. Moreover, it is more and more usual to deal with large samples taking values in high dimensional spaces. In this context, a fast recursive estimator has been introduced by Cardot et al. (2013). This work aims at stud...

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Vydáno v:Journal of multivariate analysis Ročník 146; s. 209 - 222
Hlavní autor: Godichon-Baggioni, Antoine
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier 01.04.2016
Témata:
Mathematics
Statistics
Robust statistics
Algorithms
Law of large numbers
Spatial median
Martingales in Hilbert space
Stochastic gradient
Functional data analysis
Recursive estimation
ISSN:0047-259X, 1095-7243
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Shrnutí:The geometric median, also called $L^1$-median, is often used in robust statistics. Moreover, it is more and more usual to deal with large samples taking values in high dimensional spaces. In this context, a fast recursive estimator has been introduced by Cardot et al. (2013). This work aims at studying more precisely the asymptotic behavior of the estimators of the geometric median based on such non linear stochastic gradient algorithms. The $L^p$ rates of convergence as well as almost sure rates of convergence of these estimators are derived in general separable Hilbert spaces. Moreover, the optimal rates of convergence in quadratic mean of the averaged algorithm are also given.
ISSN:0047-259X
1095-7243
DOI:10.1016/j.jmva.2015.09.013