A Class Of Nonparametric ARMA Models

A new class of nonparametric autoregressive moving average models, in which arbitrary univariate functions act as the coefficients of autoregressive terms instead of constants, is proposed and discussed. The probabilistic properties of the models are investigated and a sufficient condition of stationarity is derived firstly. The local linear and polynomial spline approaches are respectively used to estimate the the functional coefficients. In local linear estimation, the optimal bandwith is selected via a modified generalized cross-validation criterion. Correspondingly, the number of knots are determined by virtue of AIC criterion in spline method, the locations of knots are chosed by equally quantile . We apply a bootstrap method to test whether the functional coefficients are some specified parametric forms. The feasibility and validity of proposed methods are illustrated by simulated and real date exemples.