Recursive Estimation of Conditional Spatial Medians and Conditional Quantiles

摘要

We consider the problem of constructing an on-line (recursive) algorithm for tracking a conditional spatial median, a center of a multivariate distribution. In the one-dimensional case we also track conditional quantiles of arbitrary level. We establish a nonasymptotic upper bound for the L_p-risk of the algorithm, which is then minimized under different assumptions on the magnitude of the variation of the spatial median or quantile. We derive convergence rates for the examples we consider.