Statistics Surveys

Discrete variations of the fractional Brownian motion in the presence of outliers and an additive noise

Sophie Achard and Jean-François Coeurjolly

Full-text: Open access

Abstract

This paper gives an overview of the problem of estimating the Hurst parameter of a fractional Brownian motion when the data are observed with outliers and/or with an additive noise by using methods based on discrete variations. We show that the classical estimation procedure based on the log-linearity of the variogram of dilated series is made more robust to outliers and/or an additive noise by considering sample quantiles and trimmed means of the squared series or differences of empirical variances. These different procedures are compared and discussed through a large simulation study and are implemented in the R package dvfBm.

Article information

Source
Statist. Surv., Volume 4 (2010), 117-147.

Dates
First available in Project Euclid: 11 June 2010

Permanent link to this document
https://projecteuclid.org/euclid.ssu/1276260873

Digital Object Identifier
doi:10.1214/09-SS059

Mathematical Reviews number (MathSciNet)
MR2658892

Zentralblatt MATH identifier
1267.60040

Subjects
Primary: 60G15: Gaussian processes 62F10: Point estimation
Secondary: 62F35: Robustness and adaptive procedures

Keywords
Fractional Brownian motion, Hurst exponent estimation, discrete variations, robustness, outliers

Citation

Achard, Sophie; Coeurjolly, Jean-François. Discrete variations of the fractional Brownian motion in the presence of outliers and an additive noise. Statist. Surv. 4 (2010), 117--147. doi:10.1214/09-SS059. https://projecteuclid.org/euclid.ssu/1276260873


Export citation

References

  • S. Baykut, T. Akgül, and S. Ergintav. Estimation of spectral exponent parameter of 1/f process in additive white background noise., EURASIP Journal on Advances in Signal Processing, 2007(15):1–7, 2007.
  • J. Beran., Statistics for long-memory processes. Chapman & Hall/CRC, 1994.
  • A. Brouste, J. Istas, and S. Lambert-Lacroix. On Fractional Gaussian Random Fields Simulations., Journal of Statistical Software, 23(1): 1–23, 2007.
  • G. Chan and A.T.A. Wood. Estimation of fractal dimension for a class of non-Gaussian stationary processes and fields., Annals of Statistics, 32(3) :1222–1260, 2004.
  • J.-F. Coeurjolly. Simulation and identification of the fractional Brownian motion: a bibliographical and comparative study., J. Stat. Softw., 5(7):1–53, November 2000a.
  • J.-F. Coeurjolly. Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths., Stat. Infer. Stoch. Process., 4(2): 199–227, January 2001.
  • J.-F. Coeurjolly. Identification of multifractional Brownian motion., Bernoulli, 11(6):987 –1008, 2005.
  • J.-F. Coeurjolly. Hurst exponent estimation of locally self-similar gaussian processes using sample quantiles., Annals of Statistics, 36(3) :1404–1434, 2008.
  • J.-F. Coeurjolly., Inférence statistique pour les mouvements Browniens fractionnaire et multifractinonaire. PhD thesis, Université Joseph Fourier, 2000b.
  • S. Cohen and J. Istas. An universal estimator of local self-similarity., ESAIM: Probability and Statistics, 2002.
  • I. Daubechies. Orthonormal bases of compactly supported wavelets., Communications on Pure and, Applied Mathematics, 41 (7):909–996, 2006.
  • J.L. Doob., Stochastic Processes. Wiley Classics Library, USA, 1953.
  • P. Doukhan, G. Oppenheim, and M.S. Taqqu., Theory and applications of long-range dependence. Birkhauser, 2003.
  • G. Fäy, E. Moulines, F. Roueff, and M.S. Taqqu. Estimators of long-memory: Fourier versus wavelets., Journal of Econometrics, 151(2):159–177, 2009.
  • J. Istas. Quadratic variations of spherical fractional Brownian motions., Stochastic Processes and their Applications, 117 (4):476–486, 2007.
  • J. Istas and G. Lang. Quadratic variations and estimation of the hölder index of a gaussian process., Ann. Inst. H. Poincaré Probab. Statist., 33: 407–436, 1997.
  • J.T. Kent and A.T.A. Wood. Estimating the fractal dimension of a locally self-similar gaussian process using increments., J. Roy. Statist. Soc. Ser. B, 59:679–700, 1997.
  • B. Mandelbrot and J. Van Ness. Fractional brownian motions, fractional noises and applications., SIAM Rev., 10:422–437, 1968.
  • D. B. Percival and A. T. Walden., Wavelet Methods for Time Series Analysis. Cambridge University Press, 2000.
  • F. Richard and H. Biermé. Estimation of anisotropic gaussian fields through radon transform., ESAIM Probab. Stat., 12(1):30–50, 2008.
  • Haipeng Shen, Zhengyuan Zhu, and Thomas C. M. Lee. Robust estimation of the self-similarity parameter in network traffic using wavelet transform., Signal Process., 87(9) :2111–2124, 2007.