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June, 1957 On Consistent Estimates of the Spectrum of a Stationary Time Series
Emanuel Parzen
Ann. Math. Statist. 28(2): 329-348 (June, 1957). DOI: 10.1214/aoms/1177706962


This paper is concerned with the spectral analysis of wide sense stationary time series which possess a spectral density function and whose fourth moment functions satisfy an integrability condition (which includes Gaussian processes). Consistent estimates are obtained for the spectral density function as well as for the spectral distribution function and a general class of spectral averages. Optimum consistent estimates are chosen on the basis of criteria involving the notions of order of consistency and asymptotic variance. The problem of interpolating the estimated spectral density, so that only a finite number of quantities need be computed to determine the entire graph, is also discussed. Both continuous and discrete time series are treated.


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Emanuel Parzen. "On Consistent Estimates of the Spectrum of a Stationary Time Series." Ann. Math. Statist. 28 (2) 329 - 348, June, 1957.


Published: June, 1957
First available in Project Euclid: 27 April 2007

zbMATH: 0081.14102
MathSciNet: MR88833
Digital Object Identifier: 10.1214/aoms/1177706962

Rights: Copyright © 1957 Institute of Mathematical Statistics

Vol.28 • No. 2 • June, 1957
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