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A Non‐Parametric Estimator of the Spectral Density of a Continuous‐Time Gaussian Process Observed at Random Times

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  • JEAN‐MARC BARDET
  • PIERRE R. BERTRAND

Abstract

. In numerous applications data are observed at random times and an estimated graph of the spectral density may be relevant for characterizing and explaining phenomena. By using a wavelet analysis, one derives a non‐parametric estimator of the spectral density of a Gaussian process with stationary increments (or a stationary Gaussian process) from the observation of one path at random discrete times. For every positive frequency, this estimator is proved to satisfy a central limit theorem with a convergence rate depending on the roughness of the process and the moment of random durations between successive observations. In the case of stationary Gaussian processes, one can compare this estimator with estimators based on the empirical periodogram. Both estimators reach the same optimal rate of convergence, but the estimator based on wavelet analysis converges for a different class of random times. Simulation examples and an application to biological data are also provided.

Suggested Citation

  • Jean‐Marc Bardet & Pierre R. Bertrand, 2010. "A Non‐Parametric Estimator of the Spectral Density of a Continuous‐Time Gaussian Process Observed at Random Times," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(3), pages 458-476, September.
    • Handle: RePEc:bla:scjsta:v:37:y:2010:i:3:p:458-476
    DOI: 10.1111/j.1467-9469.2009.00684.x
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    References listed on IDEAS

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    1. J. Bardet & G. Lang & E. Moulines & P. Soulier, 2000. "Wavelet Estimator of Long-Range Dependent Processes," Statistical Inference for Stochastic Processes, Springer, vol. 3(1), pages 85-99, January.
    2. Lii, Keh-Shin & Masry, Elias, 1994. "Spectral estimation of continuous-time stationary processes from random sampling," Stochastic Processes and their Applications, Elsevier, vol. 52(1), pages 39-64, August.
    3. Jean‐Marc Bardet & Pierre Bertrand, 2007. "Identification of the multiscale fractional Brownian motion with biomechanical applications," Journal of Time Series Analysis, Wiley Blackwell, vol. 28(1), pages 1-52, January.
    4. E. Moulines & F. Roueff & M. S. Taqqu, 2007. "On the Spectral Density of the Wavelet Coefficients of Long‐Memory Time Series with Application to the Log‐Regression Estimation of the Memory Parameter," Journal of Time Series Analysis, Wiley Blackwell, vol. 28(2), pages 155-187, March.
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    Cited by:

    1. Lihong Wang & Jinde Wang, 2014. "Wavelet estimation of the memory parameter for long range dependent random fields," Statistical Papers, Springer, vol. 55(4), pages 1145-1158, November.
    2. Bardet, J.-M. & Tudor, C.A., 2010. "A wavelet analysis of the Rosenblatt process: Chaos expansion and estimation of the self-similarity parameter," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2331-2362, December.
    3. Richard Hunt & Shelton Peiris & Neville Weber, 2022. "Estimation methods for stationary Gegenbauer processes," Statistical Papers, Springer, vol. 63(6), pages 1707-1741, December.

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