On the integral of the squared periodogram
type: Item
instance of: scholarly article
title: On the integral of the squared periodogram
zbMATH Open document ID: 0995.62012
DOI: 10.1016/S0304-4149(99)00071-X
author: Rohit Deo
author: Willa W. Chen
published in: Stochastic Processes and their Applications
publication date: 2002-08-29T00:00:00Z
review text: Let \(X_1,\dots , X_n \) be observations of a zero mean stationary Gaussian time series with spectral density \(f\), and let \(I(\lambda)\) be the periodogram \[ I(\lambda)=(2\pi n)^{-1}\left |\sum _{t=1}^n X_t e^{-it\lambda }\right |^2 . \] For a function \(\phi \) consider the functional \[ g_n(I) = \int _{-\pi }^{\pi } \eta (\lambda)\phi (I(\lambda))d\lambda , \] where \(\eta \) is a continuous even function, and its discrete version \[ b_n(I)=2\pi n^{-1}\sum _{i=1}^{n-1} \eta (\lambda _i) \phi (I(\lambda _i)), \] where \(\lambda _i=2\pi in^{-1}, \;i = 1,\dots , {n-1}.\) Here, it is proved that if \(\{X_t\}\) is Gaussian white noise and \(\phi \) a non-linear function (especially, \(\phi (\lambda) = \lambda ^2 \) is considered), then the asymptotic variance of \(g_n(I)\) is not the same as that of \(b_n(I)\). For a general stationary Gaussian sequence, the joint asymptotic distribution of the vector \[ \left (\int _{-\pi } ^\pi I_n(\lambda)d\lambda , \int _ {-\pi }^\pi I_n^2(\lambda) d\lambda \right) \] is established, where \(I_n(\lambda)\) is the mean corrected periodogram. The result is then extended to the asymptotic distribution of a goodness-of-fit test statistic for testing hypotheses on spectral densities.
reviewed by: Zuzana Prášková
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 62M15
zbMATH DE Number: 1792496
zbMATH Keywords: periodogram
zbMATH Keywords: spectral density
zbMATH Keywords: goodness-of-fit
MaRDI profile type: Publication
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full work available at URL: s0304-4149(99)00071-x
OpenAlex ID: W2041257755
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description: scientific article; zbMATH DE number 1792496