On Asymptotically Efficient Consistent Estimates of the Spectral Density Function of a Stationary Time Series

Summary Several general classes of estimates of the spectral density function of a stationary time series are introduced, which are shown to include most estimates previously suggested by various researchers. Asymptotic expressions are given for the mean square error of these estimates. We thus determine, for each estimate, the class of time series for whose spectral density it is a consistent estimate. Further, for a given time series, the minimum integrated mean square error with which its spectral density may be estimated is determined, and evaluated for large sample sizes. Efficient estimates are defined as those which achieve, asymptotically, the minimum mean integrated square error. The asymptotic formulae derived are used to make a numerical comparison of some estimates which have been suggested. However, in this paper no conclusions are drawn or principles enunciated as to how to proceed in practice to estimate the spectral density.