Functional convergence and optimality of plug-in estimators for stationary densities of moving average processes

Abstract

We give new results, under mild assumptions, on convergence rates in L₁ and L₂ for residual-based kernel estimators of the innovation density of moving average processes. Exploiting the convolution representation of the stationary density of moving average processes, these estimators can be used to obtain n^1/2-consistent plug-in estimators for this stationary density. Here we derive functional weak convergence results in L₁ and C₀(R) for these plug-in estimators. If efficient estimators for the finite-dimensional parameters of the process are used in our construction, semiparametric efficiency of our plug-in estimators is obtained.