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)$E(h(X_{n+1})|X_1,\dots ,X_n)$. 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)$L_2(F)$, 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)$P(X_{n+1}?·|X_1,\dots ,X_n)$ 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.