On the Asymptotic Optimality of Spectral Analysis for Testing Hypotheses About Time Series

Abstract

Classification of a sample from a zero mean, stationary, Gaussian time series into populations distinguished by characteristics of the spectrum can be done with a decision theoretic procedure or spectral analysis. Decision theory requires that each population be characterized by a probability distribution on the space of spectral density functions. In this paper, we relate the two methods by showing that under many conditions, as the sample length increases, the expected cost of the Bayes test formed from spectral estimates by approximating their sampling distribution by a product of chi-squared distributions approaches the expected cost of the Bayes test formed from the original data. The amount of smoothing that can be used in the spectral estimates depends on the prior knowledge of the order of differentiability of the spectrum. This result is related to but weaker than the notion that spectral estimates are asymptotically sufficient statistics for the second order properties of a stationary Gaussian time series.