Electronic Journal of Statistics

Estimating self-similarity through complex variations

Jacques Istas

Full-text: Open access

Abstract

We estimate the self-similarity index of a $H$-sssi process through complex variations. The advantage of the complex variations is that they do not require existence of moments and can therefore be used for infinite variance processes.

Article information

Source
Electron. J. Statist., Volume 6 (2012), 1392-1408.

Dates
First available in Project Euclid: 31 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1343738543

Digital Object Identifier
doi:10.1214/12-EJS717

Mathematical Reviews number (MathSciNet)
MR2988452

Zentralblatt MATH identifier
1334.60052

Subjects
Primary: 60G18: Self-similar processes
Secondary: 60G15: Gaussian processes 60G52: Stable processes

Keywords
Self-similarity complex variations $H$-sssi processes

Citation

Istas, Jacques. Estimating self-similarity through complex variations. Electron. J. Statist. 6 (2012), 1392--1408. doi:10.1214/12-EJS717. https://projecteuclid.org/euclid.ejs/1343738543


Export citation

References

  • [1] Bardet, J.-M. and Tudor, C. (2010). A wavelet analysis of the Rosenblatt process: chaos expansion and estimation of the self-similarity parameter., Stochastic Analysis and Applications 1–25.
  • [2] Breuer, P. and Major, P. (1983). Central limit theorems for non-linear functionals of Gaussian fields., J. Mult. Anal. 13, 425-441.
  • [3] Brouste, A., Lambert-Lacroix, S. and Istas, J. (2007). On Gaussian multifractal random fields simulations., Journal of Statistical Software 1 (23), 1–23.
  • [4] Chronopoulou, A., Tudor, C. and Viens, F. (2009). Variations and Hurst index estimation for a Rosenblatt process using longer filters., Electronic Journal of Statistics 3 1393–1435.
  • [5] Chronopoulou, A., Tudor, C. and Viens, F. (2011). Self-similarity parameter estimation and reproduction property for non-Gaussian Hermite processes., Communications on Stochastic Analysis, (to appear).
  • [6] Coeurjolly, J.-F. (2001). Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths., Statistical Inference for Stochastic Processes 4 (2), 199–227.
  • [7] Cohen, S. and Istas, J. (2005). An universal estimator of local self-similarity, unpublished.
  • [8] Gradshteyn, I. and Ryzhik, I. (2007)., Table of Integrals, Series, and Products, Alan Jeffrey and Daniel Zwillinger (eds.), Seventh edition.
  • [9] Istas, J. (2011). Manifold indexed fractional fields., ESAIM P & S (to appear).
  • [10] Istas, J. and Lang, G. (1997). Quadratic variations and estimation of the Hölder index of a Gaussian process., Ann. Inst. Poincaré 33,4 407–436.
  • [11] Lacaux, C. and Loubes, J.-M. (2007). Hurst exponent estimation of fractional Lévy Motions., Alea 3 143–164.
  • [12] Nourdin, I., Nualart, D. and Tudor, C. (2010). Central and non-central limit theorems for weighted power variations of fractional Brownian motion., Ann. Inst. H. Poincaré Probab. Statist. 46,4 1055–1079.
  • [13] Nourdin, I. and Peccati, G. (2011). Normal approximations with Malliavin calculus. From Stein’s method to universality., Cambridge University Press (to appear).
  • [14] Pipiras, V., Taqqu, M. and Abry, P. (2007). Bounds for the covariance of functions of infinite variance stable random variables with applications to central limit theorems and wavelet-based estimation., Bernoulli 13,4 1091–1123.
  • [15] Samorodnitsky, G. and Taqqu, M. (1994)., Stable non-Gaussian random processes: stochastic models with infinite variance, Chapman & Hall, New York.
  • [16] Takenaka, S. (1991). Integral-geometric construction of self-similar stable processes., Nagoya Math. J. 123 1–12.
  • [17] Tudor, C. and Viens, F. (2009). Variations and estimators for the selfsimilarity order through Malliavin calculus., The Annals of Probability 37,6 2093–2134.