Statistical Analysis Based on Functionals of Spectra

Abstract

Many important quantities in stationary time series are often expressed as functionals of spectral density. For a linear functional, a natural idea of constructing an estimator is to replace an unknown spectral density by the periodogram based on the data. The functional of interest is, however, not always linear with respect to the spectral density. For the nonlinear case replacing the unknown spectral density by the periodogram causes asymptotic bias since the periodogram divided by the spectral density is asymptotically exponential. It is well known that the periodogram is an asymptotically unbiased but inconsistent estimator of the spectral density. By virtue of the fact of asymptotic independence of the periodograms, it is possible to construct a consistent estimator of the spectal density by an approach that uses a smoothed periodogram (nonparametric kernel spectral estimator). Although the rate of convergence of the nonparametric kernel spectral estimator is smaller than the usual order n ^1/2, where n is a sample size, the integration of the nonparametric kernel spectral estimator over [-π, π] leads to the recovery of the ordinary n ^1/2 asymptotics.