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Functional convergence and optimality of plug-in estimators for stationary densities of moving average processes. (English) Zbl 1058.62072

Summary: We give new results, under mild assumptions, on convergence rates in \(L_1\) and \(L_2\) 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 \(L_1\) and \(C_0(\mathbb R)\) 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.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60F17 Functional limit theorems; invariance principles
62G07 Density estimation
60F05 Central limit and other weak theorems
60F25 \(L^p\)-limit theorems
62G20 Asymptotic properties of nonparametric inference
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