Stationary linear and non-linear system identification and predictor set completeness

A very general consistency theorem for stationary nonlinear prediction error estimators is presented. Because this result does not require a representative of the system generating the observations (if such exists) to lie in the set of candidate predictors it applies to the practical problem of modelling complex systems with simple models. In order to measure the fit between observed processes and any given predictor set we introduced three notions of predictor set completeness an illustrate the definitions with some examples. The main consistency result is then re-expressed in terms of these notions and we examine their relationship to Ljung's definitions of identifiability. Amongst other results the strong consistency of maximum likelihood estimators for Gaussian auto-regressive moving average systems may be obtained as a special case of our general results (see [3,7]).