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.