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The Distribution Of Periodogram Ordinates

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  • Chen Zhao‐Guo
  • E. J. Hannan

Abstract

. The empirical distribution function of yi=I(ωj)/{2πf(ωj)}, ωj= 2πj/T, where I(ω) is the periodogram for a set of observations from a stationary time series with spectral density f(ω), is shown to converge, almost surely, to the distribution with density exp(‐ x), under appropriate conditions. The same methods are used to prove the convergence, almost surely, of an estimate of the prediction error variance constructed from the I(ωj) and of the complex empirical distribution function based on the Fourier coefficients.

Suggested Citation

  • Chen Zhao‐Guo & E. J. Hannan, 1980. "The Distribution Of Periodogram Ordinates," Journal of Time Series Analysis, Wiley Blackwell, vol. 1(1), pages 73-82, January.
    • Handle: RePEc:bla:jtsera:v:1:y:1980:i:1:p:73-82
    DOI: 10.1111/j.1467-9892.1980.tb00301.x
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    Cited by:

    1. Proietti, Tommaso & Luati, Alessandra, 2015. "The generalised autocovariance function," Journal of Econometrics, Elsevier, vol. 186(1), pages 245-257.
    2. 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.
    3. Proietti, Tommaso & Lütkepohl, Helmut, 2013. "Does the Box–Cox transformation help in forecasting macroeconomic time series?," International Journal of Forecasting, Elsevier, vol. 29(1), pages 88-99.
    4. Velasco, Carlos, 2000. "Non-Gaussian Log-Periodogram Regression," Econometric Theory, Cambridge University Press, vol. 16(1), pages 44-79, February.
    5. Daniel Janas & Rainer von Sachs, 1995. "Consistency For Non‐Linear Functions Of The Periodogram Of Tapered Data," Journal of Time Series Analysis, Wiley Blackwell, vol. 16(6), pages 585-606, November.
    6. Hassler, U. & Marmol, F. & Velasco, C., 2006. "Residual log-periodogram inference for long-run relationships," Journal of Econometrics, Elsevier, vol. 130(1), pages 165-207, January.
    7. Chang Sik Kim & Peter C.B. Phillips, 2006. "Log Periodogram Regression: The Nonstationary Case," Cowles Foundation Discussion Papers 1587, Cowles Foundation for Research in Economics, Yale University.
    8. Mahmoud El-Morshedy & Abd El-Moneim A. M. Teamah & Mohammed H. El-Menshawy & Rashad M. EL-Sagheer & Hasnaa M. Faied & Afrah Al-Bossly & Mohamed S. Eliwa, 2022. "Some Asymptotic Properties of a Kernel Bispectum Estimate with Different Multitapers," Mathematics, MDPI, vol. 10(18), pages 1-23, September.
    9. Kokoszka, Piotr & Mikosch, Thomas, 2000. "The periodogram at the Fourier frequencies," Stochastic Processes and their Applications, Elsevier, vol. 86(1), pages 49-79, March.
    10. Hernandez-Flores, C. N. & Artiles-Romero, J. & Saavedra-Santana, P., 1999. "Estimation of the population spectrum with replicated time series," Computational Statistics & Data Analysis, Elsevier, vol. 30(3), pages 271-280, May.
    11. Fay, Gilles & Soulier, Philippe, 2001. "The periodogram of an i.i.d. sequence," Stochastic Processes and their Applications, Elsevier, vol. 92(2), pages 315-343, April.

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