On robust recursive nonparametric curve estimation

The authors consider the problem of estimating a regression function $\theta\colon \ [0,1]\rightarrow{\bf R}$ when moments of the estimation errors may not exist. More specifically, $\theta$ is assumed to lie in some class $\Theta_\beta$ of Lipschitz functions with smoothness $\beta.$ The distribution of the i.i.d. error terms is assumed to have zero median and a Lipschitz continuous density that is bounded away from zero in some interval around zero. The design is assumed to be almost equidistant. For the problem, a robust recursive estimate is proposed that is based on a stochastic approximation procedure. The rate of uniform (over both $\Theta_\beta$ and a suitable subset of $[0,1]$) convergence is derived for the estimate.