## Online estimation of the geometric median in Hilbert spaces: nonasymptotic confidence balls.(English)Zbl 1371.62027

Summary: 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 the first two authors with P.-A. Zitt [Bernoulli 19, No. 1, 18–43 (2013; Zbl 1259.62068)]. 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.

### MSC:

 62G05 Nonparametric estimation 62L20 Stochastic approximation 62G15 Nonparametric tolerance and confidence regions 62G35 Nonparametric robustness

Zbl 1259.62068
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