No general analytical tools are available to estimate the state of a nonlinear stochastic system observed through a nonlinear noisy channel. This problem is addressed in this paper under the assumption that the statistics of the random variables are unknown, hence a statistical approach is followed instead of a probabilistic one. The following approximations are enforced : (i) the state estimator is a finite-memory one, (ii) the estimation functions are given fixed structures in which a certain number of parameters have to be optimized (multilayer feedforward neural networks are chosen from among various possible nonlinear approximators), (iii) the algorithms for optimizing the parameters (i.e. the network weights) rely on a stochastic approximation. Simulation results are reported to compare the behaviour of the proposed estimator with the extended Kalman filter and the estimators based on the on-line minimization of the estimation error.