ONLINE ESTIMATION OF THE GEOMETRIC MEDIAN IN HILBERT SPACES: NONASYMPTOTIC CONFIDENCE BALLS

• 成果类型: Article
• 署名作者: Cardot, Herve; Cenac, Peggy; Godichon-Baggioni, Antoine
• 署名单位: Universite Bourgogne Europe
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/16-AOS1460
• 发表日期: 2017
• 页码: 591-614
• 关键词: stochastic-approximation

摘要

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.