On the integral of the squared periodogram
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Let X1, X2.,., Xn be a sample from a stationary Gaussian time series and let I(·) be the sample periodogram. Some researchers have either proved heuristically or claimed that under general conditions, the asymptotic behaviour of ∫-ππη(λ)φ(I(λ))dλ is equivalent to that of the discrete version of the integral given by (2π/n)∑i=1n-1η(λi)φ(I(λi)), where λi are the Fourier frequencies and φ and η are suitable possibly non-linear functions. In this paper, we prove that this asymptotic equivalence is not true when φ is a non-linear function. We derive the exact finite sample variance of ∫-ππI2(λ)dλ when (Xt) is Gaussian white noise and show that it is asymptotically different from that of (2π/n)∑i=1n-1I2(λi). The asymptotic distribution of ∫-ππI2(λ)dλ is also obtained in this case. The result is then extended to obtain the limiting distribution of ∫-ππf-2(λ)I2(λ)dλ when (Xt) is a stationary Gaussian series with spectral density f(·). From these results, the limiting distribution of the integral version of a goodness-of-fit statistic proposed in the literature is obtained. © 2000 Elsevier Science B.V.
author list (cited authors)
Deo, R. S., & Chen, W. W.