Cumulant Spectral Estimates

Abstract

In Chapter V a class of spectral density estimates were introduced and their asymptotic properties were analyzed. There two alternative representations as given by (V. 18) and (V. 19) were given. In the early days of spectral analysis, it was common to use representation (V. 19) so that covariances were initially estimated and then Fourier transformed (with weights) so as to obtain spectral density estimates. In recent years, it has been more usual to employ the alternative representation (V. 18) of the spectral density estimate as a smoothed periodogram. Actually a Riemann sum as a discrete approximation to (V. 18) is considered. A discrete Fourier transform is used to compute the periodogram at the frequencies 2 π j/N, j = 0, 1, ... , N − 1, and the periodogram is then smoothed to produce the spectral density estimate. For composite numbers N there is a computational advantage in making use of the finite Fourier transform and carrying out the computation by employing the fast Fourier transform. For this reason, there will be an introductory discussion of the finite Fourier transform and fast Fourier transform in section 2.