Recursive estimation of the conditional geometric median in Hilbert spaces

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.