On estimation of a class of nonlinear systems by the kernel regression estimate

The estimation of a multiple-input single-output discrete Hammerstein system is studied. Such a system contains a nonlinear memoryless subsystem followed by a dynamic linear subsystem. The impulse response of the dynamic linear subsystem is obtained by the correlation method. The main results concern the estimation of the nonlinear memoryless subsystem. No conditions are imposed on the functional form of the nonlinear subsystem, and the nonlinearity is recovered using the kernel regression estimate. The distribution-free pointwise and global convergence of the estimate is demonstrated-that is, no conditions are imposed on the input distribution, and convergence is proven for virtually all nonlinearities. The rates of pointwise as well as global convergence are obtained for all input distributions and for Lipschitz type nonlinearities