On a spectral density estimate obtained by averaging periodograms

A spectral density statistic obtained by averaging periodograms over overlapping time intervals is considered where the periodograms are calculated using a data window. The asymptotic mean square error of this estimate for scale parameter windows is determined and, as an example, it is shown that the use of the Tukey–Hanning window leads partially to a smaller mean square error than a window suggested by Kolmogorov and Zhurbenko. Furthermore the Tukey–Hanning window is independent of the unknown spectral density, which is not the case for the Kolmogorov–Zhurbenko window. The mean square error of this estimate is also less than the mean square error of commonly used window estimates. Finally, a central limit theorem for the estimate is established.

On a spectral density estimate obtained by averaging periodograms

A spectral density statistic obtained by averaging periodograms over overlapping time intervals is considered where the periodograms are calculated using a data window. The asymptotic mean square error of this estimate for scale parameter windows is determined and, as an example, it is shown that the use of the Tukey–Hanning window leads partially to a smaller mean square error than a window suggested by Kolmogorov and Zhurbenko. Furthermore the Tukey–Hanning window is independent of the unknown spectral density, which is not the case for the Kolmogorov–Zhurbenko window. The mean square error of this estimate is also less than the mean square error of commonly used window estimates. Finally, a central limit theorem for the estimate is established.

The relationship between maximum entropy and maximum likelihood spectra

Burg (1972) established an analytical relationship between maximum entropy and maximum likelihood spectral density estimates. If [Formula: see text], j = 0, 1,… M − 1 and M successive maximum entropy spectral density estimates at frequency f, then the M‐length maximum likelihood spectral density estimate [Formula: see text] is given by [Formula: see text]. (1) The maximum entropy estimates are made via Burg's well‐known formula,[Formula: see text], (2) where [Formula: see text], k = 0, 1,…,N are the coefficients of the N‐length prediction‐error filter, [Formula: see text] is its error power, and Δt is the sample interval. These estimates can be recursively calculated as shown by Burg (Smylie et al, 1973).