Prediction in moving average processes

For the stationary invertible moving average process of order one with unknown innovation distribution F , we construct root-n consistent plug-in estimators of conditional expectations E(h(Xn+1)|X1, . . . , Xn). More specifically, we give weak conditions under which such estimators admit Bahadur type representations, assuming some smoothness of h or of F . For fixed h it suffices that h is locally of bounded variation and locally Lipschitz in L2(F ), and that the convolution of h and F is continuously differentiable. A uniform representation for the plug-in estimator of the conditional distribution function P (Xn+1 ≤ · |X1, . . . , Xn) holds if F has a uniformly continuous density. For a smoothed version of our estimator, the Bahadur representation holds uniformly over each class of functions h that have an appropriate envelope and whose shifts are F -Donsker, assuming some smoothness of F . The proofs use empirical process arguments. AMS 2000 subject classification. Primary: 62M09, 62M10.