A Note on the Behaviour of Nonparametric Density and Spectral Density Estimators at Zero Points of Their Support

Abstract

The asymptotic behaviour of nonparametric estimators of the stationary density and of the spectral density function of a stationary process have been studied in some detail in the last 50–60years. Nevertheless, less is known about the behaviour of these estimators when the target function happens to vanish at the point of interest. In the article at hand, we fill this gap and show that asymptotic normality still holds true but with super‐efficient and different rates of convergence for the density and for the spectral density estimators that are affected also by the dependence structure of the process.

Keywords: Central limit theorem, kernel smoothing, nonparametric estimation, overdifferencing, time series