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This paper considers the nonparametric estimation problem for a class of nonlinear time series systems that are characterized by their block-oriented structure. This consists of series and parallel connections where a nonlinear memoryless subsystem is imbedded in the linear dynamical model. The input-output training data generated from the system are dependent and they do not reveal the strong mixing property. The latter is a common assumption required in the nonparametric estimation theory for dependent data. The nonlinear part of the system is recovered with the weighted $k$-nearest neighbor regression estimate. The \textit{a priori} information is nonparametric, both the nonlinear characteristic and the impulse response of the linear part are completely unknown and can be of any form. Local and global properties of the estimate are examined. Whatever the probability density of the input signal, the estimate converges at every continuity point of the characteristic as well as in the global sense. The convergence rate is evaluated and it is found to be independent of the shape of the input density. These results allow us to find a set of optimal weights that further improve the accuracy of our estimation algorithm. Particularly, the weights that take into account the correlation structure of the observed data are considered. This reveals the advantage of the weighted $k$-nearest neighbor estimate over the commonly used kernel estimates. The obtained results are also extended to other types of nonparametric time series models that are driven by spike trains data.
