On the Efficiency of Spectral Density Estimates of a Stationary Process

Zhurbenko, I. G.

doi:10.1137/1125059

Cited by 8 publications

(11 citation statements)

References 8 publications

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“…This is the result obtained by Zhurbenko (1980b), Theorem 4; see also (1979a), Theorem 4 and (1980a), Theorem 3.…”

Section: The Mean Square Errorsupporting

confidence: 81%

“…Zhurbenko investigates the mean square error of fN(A) and considers a sequence aM(t) which leads to a smaller mean square error than the commonly used window-type spectral density estimates (see e.g. Zhurbenko (1980b)). He further shows that the statistic is less sensitive to disturbances from outlying frequencies than the window estimates (Zhurbenko (1983), Theorem 8 and Theorem 11, see also Zhurbenko and Kozhevnikova (1982)).…”

Section: Introductionmentioning

confidence: 99%

“…Zhurbenko (1980b)). He further shows that the statistic is less sensitive to disturbances from outlying frequencies than the window estimates (Zhurbenko (1983), Theorem 8 and Theorem 11, see also Zhurbenko and Kozhevnikova (1982)). However, it should be mentioned that he compares the shift estimate only with window estimates without data windows (Zhurbenko (1983), Theorem 8), while data windows also lead for window estimates to a substantial reduction of the influence of outlying frequencies (see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…He minimized the mean square error over this class of sequences aM(t) (e.g. Zhurbenko (1980b), Theorem 5), and compared this mean square error with the least possible mean square error and the mean square error of several commonly used window estimates (Zhurbenko (1980b), Table 1).…”

Section: Introductionmentioning

confidence: 99%

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On a spectral density estimate obtained by averaging periodograms

Dahlhaus

1985

Journal of Applied Probability

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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.

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“…This is the result obtained by Zhurbenko (1980b), Theorem 4; see also (1979a), Theorem 4 and (1980a), Theorem 3.…”

Section: The Mean Square Errorsupporting

confidence: 81%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On a spectral density estimate obtained by averaging periodograms

Dahlhaus

1985

Journal of Applied Probability

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“…For smooth spectral densities this estimate has roughly the same mean 162 R. Dahlhaus sequare error as the window estimate (with global bandwith). Zhurbenko (1980) proves for a particular taper ht, N and Lipschitz-continuous spectral densities that the mean square error of the estimate is lower than the mean square error of the window estimate with several common windows but higher than the optimal window estimate. Furthermore, Zhurbenko (1983) shows by considering a spectral measure with a jump that the segment estimate is less sensitive to disturbances from outlying frequencies than the window estimate (Zhurbenko, 1983, Theorem 8 and Theorem 11).…”

Section: T= Lm+ N-lmentioning

confidence: 94%

Nonparametric high resolution spectral estimation

Dahlhaus

1990

Probab. Th. Rel. Fields

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Summary.The uniform rate of convergence of the integrated relative mean square error over a (with the sample size T) increasing class X T of stationary processes is studied for several estimates of the spectral density. The class X r is chosen in a way such that estimates with a good uniform rate of convergence over 5~ r may be termed 'high resolution spectral estimates'. By using this criterion several effects are explained theoretically, for example the leakage effect. The advantages uf using data tapers are proved and the use of local and global bandwiths are studied. Furthermore, the behavior of segment estimates are studied. Simulations are presented for the illustration of some effects.

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Debiasing Welch’s method for spectral density estimation

Astfalck,

Sykulski,

Cripps

2024

Biometrika

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Summary Welch’s method provides an estimator of the power spectral density that is statistically consistent. This is achieved by averaging over periodograms calculated from overlapping segments of a time series. For a finite length time series, while the variance of the estimator decreases as the number of segments increase, the magnitude of the estimator’s bias increases: a bias-variance trade-off ensues when setting the segment number. We address this issue by providing a novel method for debiasing Welch’s method which maintains the computational complexity and asymptotic consistency, and leads to improved finite-sample performance. Theoretical results are given for fourth-order stationary processes with finite fourth-order moments and absolutely convergent fourth-order cumulant function. The significant bias reduction is demonstrated with numerical simulation and an application to real-world data. Our estimator also permits irregular spacing over frequency and we demonstrate how this may be employed for signal compression and further variance reduction. Code accompanying this work is available in R and python.

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On the Efficiency of Spectral Density Estimates of a Stationary Process

Cited by 8 publications

References 8 publications

On a spectral density estimate obtained by averaging periodograms

On a spectral density estimate obtained by averaging periodograms

Nonparametric high resolution spectral estimation

Debiasing Welch’s method for spectral density estimation

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