Plug-in estimators for higher-order transition densities in autoregression

In this paper we obtain root-n consistency and functional central limit theorems in weighted L1-spaces for plug-in estimators of the two-step transition density in the classical stationary linear autoregressive model of order one, assuming essentially only that the innovation density has bounded variation. We also show that plugging in a properly weighted residual-based kernel estimator for the unknown innovation density improves on plugging in an unweighted residual-based kernel estimator. These weights are chosen to exploit the fact that the innovations have mean zero. If an efficient estimator for the autoregression parameter is used, then the weighted plug-in estimator for the two-step transition density is efficient. Our approach generalizes to invertible linear processes