ON THE BEHAVIOR OF NONPARAMETRIC DENSITY AND SPECTRAL DENSITY ESTIMATORS AT ZERO POINTS OF THEIR SUPPORT

The asymptotic behavior of nonparametric estimators of the probability density function of an i.i.d. sample and of the spectral density function of a stationary time series have been studied in some detail in the last 50-60 years. Nevertheless, an open problem remains to date, namely the behavior of the estimator when the target function happens to vanish at the point of interest. In the paper at hand we fill this gap, and show that asymptotic normality still holds true but with a super-efficient rate of convergence. We also provide two possible applications where these new results can be found useful in practice.