The Exact Error in Estimating the Spectral Density at the Origin

This paper derives expressions for the exact bias and variance of a general class of spectral density estimators at the zero frequency, building on the work of Neave (The exact error in spectrum estimates. Ann. Math. Statist. 42 (1971), 961–75) who studied the case where the mean of the series is assumed known. These expressions are evaluated for 15 different windows and for a wide variety of stationary time series. The exact error of the estimators is found to depend on whether the sample mean has to be estimated, and some windows are noticeably inferior at certain values of the bandwidth. A response surface analysis reveals that the finite sample relationships between the bandwidth and the exact error are quite different from the ones suggested by asymptotic theory.