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 |
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| Jazyk: | angličtina |
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Elsevier
01.04.2016
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| ISSN: | 0047-259X, 1095-7243 |
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Godichon-Baggioni, Antoine |
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| Cites_doi | 10.1073/pnas.97.4.1423 10.3150/11-BEJ390 10.1016/j.jspi.2013.04.002 10.1214/14-AOS1226 10.1016/j.csda.2011.11.019 10.1214/aos/1176348662 10.1093/biomet/35.3-4.414 10.1016/j.spa.2011.12.011 10.1093/biomet/asn031 10.1137/S0363012998308169 10.1112/S1461157020090531 10.1214/11-AOS923 10.1007/BF01584648 10.1214/aoms/1177729586 10.1137/0330046 10.1080/02331880108802751 |
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| Keywords | Robust statistics Algorithms Law of large numbers Spatial median Martingales in Hilbert space Stochastic gradient Functional data analysis Recursive estimation |
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