Inference for the Spectrum of a Stationary Process

Abstract

In this chapter we consider problems of statistical inference for time series based on frequency-domain properties of the series. The fundamental tool used is the periodogram, which is defined in Section 10.1 for any time series {x 1,..., x n }. Section 10.2 deals with statistical tests for the presence of “hidden periodicities” in the data. Several tests are discussed, corresponding to various different models and hypotheses which we may wish to test. Spectral analysis for stationary time series, and in particular the estimation of the spectral density, depends very heavily on the asymptotic distribution as n → ∞ of the periodogram ordinates of the series {X 1,..., X n }. The essential results are contained in Theorem 10.3.2. Under rather general conditions, the periodogram ordinates I n (λ i ) at any set of frequencies λ 1,..., λ m , 0 < λ 1 < ⋯ < λ m