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On wavelet analysis of the nth order fractional Brownian motion


  • Hedi Kortas


  • Zouhaier Dhifaoui
  • Samir Ben Ammou


In this paper, we investigate the use of wavelet techniques in the study of the nth order fractional Brownian motion (n-fBm). First, we exploit the continuous wavelet transform’s capabilities in derivative calculation to construct a two-step estimator of the scaling exponent of the n-fBm process. We show, via simulation, that the proposed method improves the estimation performance of the n-fBm signals contaminated by large-scale noise. Second, we analyze the statistical properties of the n-fBm process in the time-scale plan. We demonstrate that, for a convenient choice of the wavelet basis, the discrete wavelet detail coefficients of the n-fBm process are stationary at each resolution level whereas their variance exhibits a power-law behavior. Using the latter property, we discuss a weighted least squares regression based-estimator for this class of stochastic process. Experiments carried out on simulated and real-world datasets prove the relevance of the proposed method. Copyright Springer-Verlag 2012

Suggested Citation

  • Hedi Kortas & Zouhaier Dhifaoui & Samir Ben Ammou, 2012. "On wavelet analysis of the nth order fractional Brownian motion," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 21(3), pages 251-277, August.
  • Handle: RePEc:spr:stmapp:v:21:y:2012:i:3:p:251-277
    DOI: 10.1007/s10260-012-0187-2

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    References listed on IDEAS

    1. Power, Gabriel J. & Turvey, Calum G., 2010. "Long-range dependence in the volatility of commodity futures prices: Wavelet-based evidence," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(1), pages 79-90.
    2. Dickey, David A & Fuller, Wayne A, 1981. "Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root," Econometrica, Econometric Society, vol. 49(4), pages 1057-1072, June.
    3. Pérez, D.G. & Zunino, L. & Garavaglia, M. & Rosso, O.A., 2006. "Wavelet entropy and fractional Brownian motion time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 365(2), pages 282-288.
    4. Brouste, Alexandre & Istas, Jacques & Lambert-Lacroix, Sophie, 2007. "On Fractional Gaussian Random Fields Simulations," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 23(i01).
    5. Jean-François Coeurjolly, 2001. "Estimating the Parameters of a Fractional Brownian Motion by Discrete Variations of its Sample Paths," Statistical Inference for Stochastic Processes, Springer, vol. 4(2), pages 199-227, May.
    6. Anyssa Trimech & Hedi Kortas & Salwa Benammou & Samir Benammou, 2009. "Multiscale Fama-French model: application to the French market," Journal of Risk Finance, Emerald Group Publishing, vol. 10(2), pages 179-192, February.
    7. Mielniczuk, J. & Wojdyllo, P., 2007. "Estimation of Hurst exponent revisited," Computational Statistics & Data Analysis, Elsevier, vol. 51(9), pages 4510-4525, May.
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