Open Access
October, 1966 Spectral Estimates Using Nonlinear Functions
Eugene R. Rodemich
Ann. Math. Statist. 37(5): 1237-1256 (October, 1966). DOI: 10.1214/aoms/1177699268


The main result proved here (Theorem 3) is a generalization of a formula of Goldstein [2], who showed that if the estimate $\hat S(\omega)$ for the spectral density is computed by the use of the function $y(x) = \operatorname{sgn} (x)$, and the spectrum is flat, then the dominant term in the variance of $\hat S(\omega)$ is $\frac{1}{2}\pi^2K/N$. Theorem 3 evaluates this term for nonflat spectra and for more general functions $y(x)$. This analysis shows that the loss in accuracy caused by working with $y(x)$ instead of $x$ itself can be decreased considerably by using for $y(x)$ a step function with more than two values. Some results on Gaussian process, interesting in their own right, are proved along the way.


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Eugene R. Rodemich. "Spectral Estimates Using Nonlinear Functions." Ann. Math. Statist. 37 (5) 1237 - 1256, October, 1966.


Published: October, 1966
First available in Project Euclid: 27 April 2007

zbMATH: 0149.14801
MathSciNet: MR203895
Digital Object Identifier: 10.1214/aoms/1177699268

Rights: Copyright © 1966 Institute of Mathematical Statistics

Vol.37 • No. 5 • October, 1966
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