Time Series: Theory and Methods pp 320-390 | Cite as

# 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 } < π, are asymptotically independent exponential random variables with means 2π*f*(λ_{i}), were *f* is the spectral density of {*X* _{ t }}. Consequently the periodogram *I* _{ n } is not a consistent estimator of 2π*f*. Consistent estimators can however be constructed by applying linear smoothing filters to the periodogram. The asymptotic behaviour of the resulting discrete spectral average estimators can be derived from the asymptotic behaviour of the periodogram as shown in Section 10.4.

## Keywords

Spectral Density Discrete Fourier Transform Maximum Likelihood Estimator Spectral Estimate Spectral Window## Preview

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