Nonparametric Recursive Estimation in Nonlinear ARX Models

Consider the general ${\rm ARX}(k,g)$ nonlinear process defined by the recurrence relation $y_n=f(y_{n-1},\dots,y_{n-k},x_n,\dots,x_{n-q+1})+\zeta_n$, where $\{x_n\}$, $\{\zeta_n\}$ are sequences of independent identically distributed random variables. We propose a recursive nonparametric estimator of the function $f$ and we prove its strong consistency under general assumptions on the model. We study the model properties guaranteeing that these assumptions are satisfied.