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On the integral of the squared periodogram

Author

Listed:
  • Deo, Rohit S.
  • Chen, Willa W.

Abstract

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 is equivalent to that of the discrete version of the integral given by , where [lambda]i are the Fourier frequencies and [phi] and [eta] are suitable possibly non-linear functions. In this paper, we prove that this asymptotic equivalence is not true when [phi] is a non-linear function. We derive the exact finite sample variance of when {Xt} is Gaussian white noise and show that it is asymptotically different from that of . The asymptotic distribution of is also obtained in this case. The result is then extended to obtain the limiting distribution of 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.

Suggested Citation

  • Deo, Rohit S. & Chen, Willa W., 2000. "On the integral of the squared periodogram," Stochastic Processes and their Applications, Elsevier, vol. 85(1), pages 159-176, January.
  • Handle: RePEc:eee:spapps:v:85:y:2000:i:1:p:159-176
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    References listed on IDEAS

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    1. Redmond, C. & Yukich, J. E., 1996. "Asymptotics for Euclidean functionals with power-weighted edges," Stochastic Processes and their Applications, Elsevier, vol. 61(2), pages 289-304, February.
    2. Lee, Sungchul, 1999. "Asymptotics of power-weighted Euclidean functionals," Stochastic Processes and their Applications, Elsevier, vol. 79(1), pages 109-116, January.
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    Cited by:

    1. Jennifer Brown & Les Oxley & William Rea & Marco Reale, 2008. "The Empirical Properties of Some Popular Estimators of Long Memory Processes," Working Papers in Economics 08/13, University of Canterbury, Department of Economics and Finance.
    2. Carlos Velasco & Ignacio N. Lobato, 2004. "A simple and general test for white noise," Econometric Society 2004 Latin American Meetings 112, Econometric Society.
    3. Rea, William & Oxley, Les & Reale, Marco & Brown, Jennifer, 2013. "Not all estimators are born equal: The empirical properties of some estimators of long memory," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 93(C), pages 29-42.
    4. Faÿ, Gilles, 2010. "Moment bounds for non-linear functionals of the periodogram," Stochastic Processes and their Applications, Elsevier, vol. 120(6), pages 983-1009, June.
    5. McElroy, Tucker & Holan, Scott, 2009. "A local spectral approach for assessing time series model misspecification," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 604-621, April.
    6. Soulier, Philippe, 2001. "Moment bounds and central limit theorem for functions of Gaussian vectors," Statistics & Probability Letters, Elsevier, vol. 54(2), pages 193-203, September.

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