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

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  • Godichon-Baggioni, Antoine

Abstract

The geometric median, also called L1-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 Lp 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.

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  • Godichon-Baggioni, Antoine, 2016. "Estimating the geometric median in Hilbert spaces with stochastic gradient algorithms: Lp and almost sure rates of convergence," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 209-222.
    • Handle: RePEc:eee:jmvana:v:146:y:2016:i:c:p:209-222
    DOI: 10.1016/j.jmva.2015.09.013
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    References listed on IDEAS

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    1. Arnaudon, Marc & Dombry, Clément & Phan, Anthony & Yang, Le, 2012. "Stochastic algorithms for computing means of probability measures," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1437-1455.
    2. Daniel Gervini, 2008. "Robust functional estimation using the median and spherical principal components," Biometrika, Biometrika Trust, vol. 95(3), pages 587-600.
    3. Cardot, Hervé & Cénac, Peggy & Monnez, Jean-Marie, 2012. "A fast and recursive algorithm for clustering large datasets with k-medians," Computational Statistics & Data Analysis, Elsevier, vol. 56(6), pages 1434-1449.
    4. Tsang, Eric W. K., 2014. "Old and New," Management and Organization Review, Cambridge University Press, vol. 10(03), pages 390-390, November.
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    Cited by:

    1. Godichon-Baggioni, Antoine & Lu, Wei, 2024. "Online stochastic Newton methods for estimating the geometric median and applications," Journal of Multivariate Analysis, Elsevier, vol. 202(C).
    2. Hervé Cardot & Antoine Godichon-Baggioni, 2017. "Fast estimation of the median covariation matrix with application to online robust principal components analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(3), pages 461-480, September.
    3. Cheng, Guanghui & Xiong, Qiang & Lin, Ruitao, 2024. "Online bootstrap inference for the geometric median," Computational Statistics & Data Analysis, Elsevier, vol. 197(C).
    4. Mokkadem, Abdelkader & Pelletier, Mariane, 2020. "Online estimation of integrated squared density derivatives," Statistics & Probability Letters, Elsevier, vol. 166(C).
    5. Gadat, Sébastien & Panloup, Fabien, 2023. "Optimal non-asymptotic analysis of the Ruppert–Polyak averaging stochastic algorithm," Stochastic Processes and their Applications, Elsevier, vol. 156(C), pages 312-348.
    6. Mokkadem, Abdelkader & Pelletier, Mariane, 2021. "A compact law of the iterated logarithm for online estimator of hazard rate under random censoring," Statistics & Probability Letters, Elsevier, vol. 178(C).
    7. Vishwajit Hegde & Arvind S. Menon & L. A. Prashanth & Krishna Jagannathan, 2021. "Online Estimation and Optimization of Utility-Based Shortfall Risk," Papers 2111.08805, arXiv.org, revised Nov 2023.

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