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

JOURNAL OF MULTIVARIATE ANALYSIS

卷:146

Article

Godichon-Baggioni, Antoine¹ 

[1] Univ Bourgogne, Inst Math Bourgogne, 9 Rue Alain Savary, F-21078 Dijon, France

关键词: Functional data analysis;    Law of large numbers;    Martingales in Hilbert space;    Recursive estimation;    Robust statistics;    Spatial median;    Stochastic gradient algorithms;   

DOI  :  10.1016/j.jmva.2015.09.013

来源: Elsevier

【 摘 要 】

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. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free