Spectrum estimation and harmonic analysis

TLDR

Bias control and consistency dominate the choice of spectral estimators for stationary time series from finite samples. The paper introduces a new spectral estimation method using a local eigenexpansion that solves an integral equation. Computationally, the method equals a weighted average of direct‑spectrum estimates from orthogonal data windows (discrete prolate spheroidal sequences), addressing bias and smoothing. The estimator is window‑free, consistent, effective for small samples, provides an ANOVA test for line components, achieves high resolution, relates to maximum‑likelihood estimates, and extends to coherence and polyspectrum estimation.

Abstract

In the choice of an estimator for the spectrum of a stationary time series from a finite sample of the process, the problems of bias control and consistency, or "smoothing," are dominant. In this paper we present a new method based on a "local" eigenexpansion to estimate the spectrum in terms of the solution of an integral equation. Computationally this method is equivalent to using the weishted average of a series of direct-spectrum estimates based on orthogonal data windows (discrete prolate spheroidal sequences) to treat both the bias and smoothing problems. Some of the attractive features of this estimate are: there are no arbitrary windows; it is a small sample theory; it is consistent; it provides an analysis-of-variance test for line components; and it has high resolution. We also show relations of this estimate to maximum-likelihood estimates, show that the estimation capacity of the estimate is high, and show applications to coherence and polyspectrum estimates.