The theory and procedures for the determination of an optimumpth-order nonlinear. time-invariant, model of a given system is developed in this article. The model determined is optimum in the sense that, with the input being from a Gaussian process, the mean-square difference between its output and that of the given system is the smallest possible. An analysis of the mean-square error also is presented. In addition, a Parseval theorem for nonlinear systems is derived from which a measure ofpth-order non-linearity for a given system is obtained.