On robust recursive nonparametric curve estimation

Suppose we observe X-k = theta (x(k)) + xi (k) The function theta: [0, 1] --> R, is assumed to belong a priori to a given nonparametric smoothness class, the xi (k)'s are independent identically distributed random variables with zero medians. The only prior information about the distribution of the noise is that it belongs to a rather wide class. The assumptions describing this class include cases in which no moments of the noises exist, so that linear estimation methods (for example, kernel methods) can not be applied. We propose a robust estimator based on a stochastic approximation procedure and derive its rate of convergence, as the frequency of observations n tends to infinity, in almost sure as well as in mean square sense, uniformly over the smoothness class. Finally, we discuss a multivariate formulation of the problem, a robust nonparametric M-estimator (the least deviations estimator), the so called penalized estimator, and the case when the noises are not necessarily identically distributed.