Electronic Journal of Statistics

Recursive estimation of the conditional geometric median in Hilbert spaces

Hervé Cardot, Peggy Cénac, and Pierre-André Zitt

Full-text: Open access

Abstract

A recursive estimator of the conditional geometric median in Hilbert spaces is studied. It is based on a stochastic gradient algorithm whose aim is to minimize a weighted $L_{1}$ criterion and is consequently well adapted for robust online estimation. The weights are controlled by a kernel function and an associated bandwidth. Almost sure convergence and $L^{2}$ rates of convergence are proved under general conditions on the conditional distribution as well as the sequence of descent steps of the algorithm and the sequence of bandwidths. Asymptotic normality is also proved for the averaged version of the algorithm with an optimal rate of convergence. A simulation study confirms the interest of this new and fast algorithm when the sample sizes are large. Finally, the ability of these recursive algorithms to deal with very high-dimensional data is illustrated on the robust estimation of television audience profiles conditional on the total time spent watching television over a period of 24 hours.

Article information

Source
Electron. J. Statist., Volume 6 (2012), 2535-2562.

Dates
First available in Project Euclid: 4 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1357307949

Digital Object Identifier
doi:10.1214/12-EJS759

Mathematical Reviews number (MathSciNet)
MR3020275

Zentralblatt MATH identifier
1295.62080

Subjects
Primary: 62L20: Stochastic approximation
Secondary: 60F05: Central limit and other weak theorems

Keywords
Robbins–Monro asymptotic normality averaging central limit theorem kernel regression Mallows–Wasserstein distance online data robust estimator sequential estimation stochastic gradient

Citation

Cardot, Hervé; Cénac, Peggy; Zitt, Pierre-André. Recursive estimation of the conditional geometric median in Hilbert spaces. Electron. J. Statist. 6 (2012), 2535--2562. doi:10.1214/12-EJS759. https://projecteuclid.org/euclid.ejs/1357307949


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