Lii, K. S.; Rosenblatt, M. Prolate spheroidal spectral estimates. (English) Zbl 1144.62079 Stat. Probab. Lett. 78, No. 11, 1339-1348 (2008). Summary: An estimate of the spectral density of a stationary time series can be obtained by taking the finite Fourier transform of an observed sequence \(x_{0},x_{1},\ldots ,x_{N - 1}\) of sample size \(N\) with taper a discrete prolate spheroidal sequence and computing its square modulus. It is typical to take the average \(K\) of several such estimates corresponding to different prolate spheroidal sequences with the same bandwidth \(W(N)\) as the final computed estimate. For the mean square error of such an estimate to converge to zero as \(N\rightarrow \infty \), it is shown that it is necessary to have \(W(N)\downarrow 0\) with \(NW(N)\rightarrow \infty \) as \(N\rightarrow \infty \) and significantly have \(K(N)\leq 2NW(N)\) but \(K=K(N)\rightarrow \infty \) as \(N\rightarrow \infty \). Cited in 3 Documents MSC: 62M15 Inference from stochastic processes and spectral analysis 62G07 Density estimation 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62G20 Asymptotic properties of nonparametric inference × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Kac, M.; Murdock, W.; Szegö, G., On the eigenvalues of certain hermitian forms, J. Raf. Mech. Anal., 2, 767-800 (1953) · Zbl 0051.30302 [2] Perceival, D.; Walden, A., Spectral Analysis for Physical Applications (1993), Cambridge University Press · Zbl 0796.62077 [3] Riedel, K. S.; Sidorenko, A., Minimum bias multiple taper spectral estimation, IEEE Trans. Signal Process., 43, 1, 188-195 (1995) [4] Slepian, D., Prolate spheroidal wave functions, Fourier analysis, and uncertainty—V: The discrete case, Bell System Tech. J., 57, 5, 1371-1430 (1978) · Zbl 0378.33006 [5] Riesz, F.; Sz. Nagy, B., Functional Analysis (1990), Dover · Zbl 0732.47001 [6] Thomson, D. J., Spectrum estimation and harmonic analysis, Proc. IEEE, 1055-1096 (1982) [7] Thomson, D. J., Time series analysis of Holocene climate data, Philos. Trans. R. Soc. Lond., A330, 601-616 (1990) [8] Walden, A. T.; McCoy, E. J.; Percival, D. B., The effective bandwidth of a multitaper spectral estimator, Biometrika, 82, 1, 201-214 (1995) · Zbl 0823.62076 [9] Widom, H., Lectures on Integral Equations (1969), Van Nostrand · Zbl 0181.12101 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.