Optimal and robust kernel algorithms for passive stochastic approximation

The problem of estimating a root of an equation f(x )=0 is considered in the situation where the values of f( x) are measured with random errors at random points and the choice of these points cannot be controlled. Nonlinear modification of the recursive Hardle-Nixdorf method is studied. Almost sure and mean square convergence is proved, and the rate of convergence is estimated. The optimal choice of parameters and of a kernel is presented; it is shown that for the optimal procedure the lower bound for the accuracy of arbitrary methods of solving the problem is attained