Spectral Estimates Using Nonlinear Functions

Abstract

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.