Wavelet estimation of the long memory parameter for Hermite polynomial of gaussian processes

M. Clausel; F. Roueff; M. S. Taqqu; C. Tudor

ESAIM: Probability and Statistics (2014)

  • Volume: 18, page 42-76
  • ISSN: 1292-8100

Abstract

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We consider stationary processes with long memory which are non-Gaussian and represented as Hermite polynomials of a Gaussian process. We focus on the corresponding wavelet coefficients and study the asymptotic behavior of the sum of their squares since this sum is often used for estimating the long–memory parameter. We show that the limit is not Gaussian but can be expressed using the non-Gaussian Rosenblatt process defined as a Wiener–Itô integral of order 2. This happens even if the original process is defined through a Hermite polynomial of order higher than 2.

How to cite

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Clausel, M., et al. "Wavelet estimation of the long memory parameter for Hermite polynomial of gaussian processes." ESAIM: Probability and Statistics 18 (2014): 42-76. <http://eudml.org/doc/273645>.

@article{Clausel2014,
abstract = {We consider stationary processes with long memory which are non-Gaussian and represented as Hermite polynomials of a Gaussian process. We focus on the corresponding wavelet coefficients and study the asymptotic behavior of the sum of their squares since this sum is often used for estimating the long–memory parameter. We show that the limit is not Gaussian but can be expressed using the non-Gaussian Rosenblatt process defined as a Wiener–Itô integral of order 2. This happens even if the original process is defined through a Hermite polynomial of order higher than 2.},
author = {Clausel, M., Roueff, F., Taqqu, M. S., Tudor, C.},
journal = {ESAIM: Probability and Statistics},
keywords = {Hermite processes; wavelet coefficients; wiener chaos; self-similar processes; long–range dependence; wavelet analysis; non-Gaussian stochastic processes; scalogram; long memory parameter; Wiener chaos; Hermite polynomial; Rosenblatt process},
language = {eng},
pages = {42-76},
publisher = {EDP-Sciences},
title = {Wavelet estimation of the long memory parameter for Hermite polynomial of gaussian processes},
url = {http://eudml.org/doc/273645},
volume = {18},
year = {2014},
}

TY - JOUR
AU - Clausel, M.
AU - Roueff, F.
AU - Taqqu, M. S.
AU - Tudor, C.
TI - Wavelet estimation of the long memory parameter for Hermite polynomial of gaussian processes
JO - ESAIM: Probability and Statistics
PY - 2014
PB - EDP-Sciences
VL - 18
SP - 42
EP - 76
AB - We consider stationary processes with long memory which are non-Gaussian and represented as Hermite polynomials of a Gaussian process. We focus on the corresponding wavelet coefficients and study the asymptotic behavior of the sum of their squares since this sum is often used for estimating the long–memory parameter. We show that the limit is not Gaussian but can be expressed using the non-Gaussian Rosenblatt process defined as a Wiener–Itô integral of order 2. This happens even if the original process is defined through a Hermite polynomial of order higher than 2.
LA - eng
KW - Hermite processes; wavelet coefficients; wiener chaos; self-similar processes; long–range dependence; wavelet analysis; non-Gaussian stochastic processes; scalogram; long memory parameter; Wiener chaos; Hermite polynomial; Rosenblatt process
UR - http://eudml.org/doc/273645
ER -

References

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  1. [1] P. Abry and V. Pipiras, Wavelet-based synthesis of the Rosenblatt process. Eurasip Signal Processing86 (2006) 2326–2339. Zbl1172.94348
  2. [2] P. Abry and D. Veitch, Wavelet analysis of long–range-dependent traffic. IEEE Trans. Inform. Theory44 (1998) 2–15. Zbl0905.94006MR1486645
  3. [3] P. Abry, D. Veitch and P. Flandrin, Long-range dependence: revisiting aggregation with wavelets. J. Time Ser. Anal. 19 (1998) 253–266. ISSN 0143-9782. Zbl0910.62080MR1626734
  4. [4] P. Abry, Helgason H. and V. Pipiras, Wavelet-based analysis of non-Gaussian long–range dependent processes and estimation of the Hurst parameter. Lithuanian Math. J.51 (2011) 287–302. Zbl1294.62185MR2832225
  5. [5] J.-M. Bardet, Statistical study of the wavelet analysis of fractional Brownian motion. IEEE Trans. Inform. Theory48 (2002) 991–999. Zbl1061.60036MR1908463
  6. [6] J.-M. Bardet and C.A. Tudor, A wavelet analysis of the Rosenblatt process: chaos expansion and estimation of the self-similarity parameter. Stochastic Process. Appl.120 (2010) 2331–2362. Zbl1203.60043MR2728168
  7. [7] J.-M. Bardet, G. Lang, E. Moulines and P. Soulier, Wavelet estimator of long–range dependent processes. 19th “Rencontres Franco-Belges de Statisticiens” (Marseille, 1998). Stat. Inference Stoch. Process. 3 (2000) 85–99. Zbl1054.62579MR1819288
  8. [8] J.M. Bardet, H. Bibi and A. Jouini, Adaptive wavelet based estimator of the memory parameter for stationary gaussian processes. Bernoulli14 (2008) 691–724. Zbl1155.62060MR2537808
  9. [9] J.-C. Breton and I. Nourdin, Error bounds on the non-normal approximation of hermite power variations of fractional brownian motion. Electron. Commun. Probab.13 (2008) 482–493. Zbl1189.60084MR2447835
  10. [10] A. Chronopoulou, C. Tudor and F. Viens, Self-similarity parameter estimation and reproduction property for non-gaussian Hermite processes. Commun. Stoch. Anal.5 (2011) 161–185. Zbl1331.62098MR2808541
  11. [11] M. Clausel, F. Roueff, M.S. Taqqu and C. Tudor, Large scale behavior of wavelet coefficients of non-linear subordinated processes with long memory. Appl. Comput. Harmonic Anal.32 (2012) 223–241. Zbl1244.60044MR2880280
  12. [12] M. Clausel, F. Roueff, M.S. Taqqu and C. Tudor, High order chaotic limits of wavelet scalograms under long–range dependence. Technical report, Hal–Institut Telecom (2012). http://hal-institut-telecom.archives-ouvertes.fr/hal-00662317. Zbl1280.42031MR3151747
  13. [13] R.L. Dobrushin and P. Major, Non-central limit theorems for nonlinear functionals of Gaussian fields. Z. Wahrsch. Verw. Gebiete50 (1979) 27–52. Zbl0397.60034MR550122
  14. [14] P. Embrechts and M. Maejima, Selfsimilar processes. Princeton University Press, Princeton, New York (2002). Zbl1008.60003MR1920153
  15. [15] P. Flandrin, On the spectrum of fractional Brownian motions. IEEE Trans. Inform. Theory IT-35 (1989) 197–199. MR995341
  16. [16] P. Flandrin, Some aspects of nonstationary signal processing with emphasis on time-frequency and time-scale methods. Edited by J.M. Combes, A. Grossman and Ph. Tchamitchian, Wavelets. Springer-Verlag (1989) 68–98. Zbl0825.94095MR1010899
  17. [17] P. Flandrin, Fractional Brownian motion and wavelets. Edited by M. Farge, J.C.R. Hung and J.C. Vassilicos, Fractals and Fourier Transforms-New Developments and New Applications. Oxford University Press (1991). Zbl0826.60032MR1265966
  18. [18] P. Flandrin, Time-Frequency/Time-scale Analysis, 1st edition. Academic Press (1999). Zbl0954.94003MR1681043
  19. [19] R. Fox and M.S. Taqqu. Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann. Statist.14 (1986) 517–532. Zbl0606.62096MR840512
  20. [20] L. Giraitis and D. Surgailis, Central limit theorems and other limit theorems for functionals of gaussian processes. Z. Wahrsch. verw. Gebiete 70 (1985) 191–212. Zbl0575.60024MR799146
  21. [21] L. Giraitis and M.S. Taqqu, Whittle estimator for finite-variance non-gaussian time series with long memory. Ann. Statist.27 (1999) 178–203. Zbl0945.62085MR1701107
  22. [22] A.J. Lawrance and N.T. Kottegoda, Stochastic modelling of riverflow time series. J. Roy. Statist. Soc. Ser. A140 (1977) 1–47. 
  23. [23] P. Major, Multiple Wiener-Itô integrals, vol. 849 of Lect. Notes Math. Springer, Berlin (1981). Zbl0451.60002MR611334
  24. [24] E. Moulines, F. Roueff and M.S. Taqqu, On the spectral density of the wavelet coefficients of long memory time series with application to the log-regression estimation of the memory parameter. J. Time Ser. Anal.28 (2007) 155–187. Zbl1150.62058MR2345656
  25. [25] I. Nourdin and G. Peccati, Stein’s method meets Malliavin calculus: a short survey with new estimates. Technical report, Recent Advances in Stochastic Dynamics and Stochastic Analysis 8 (2010) 207–236. Zbl1203.60065MR2807823
  26. [26] I. Nourdin and G. Peccati, Stein’s method on wiener chaos. Probability Theory and Related Fields154 (2009) 75–118. Zbl1175.60053MR2520122
  27. [27] D. Nualart, The Malliavin Calculus and Related Topics. Springer (2006). Zbl1099.60003MR2200233
  28. [28] P.M. Robinson, Log-periodogram regression of time series with long range dependence. Ann. Statist.23 (1995) 1048–1072. Zbl0838.62085MR1345214
  29. [29] P.M. Robinson, Gaussian semiparametric estimation of long range dependence. Ann. Statist.23 (1995) 1630–1661. Zbl0843.62092MR1370301
  30. [30] F. Roueff and M. S. Taqqu, Central limit theorems for arrays of decimated linear processes. Stoch. Proc. Appl.119 (2009) 3006–3041. Zbl1173.60311MR2554037
  31. [31] F. Roueff and M.S. Taqqu, Asymptotic normality of wavelet estimators of the memory parameter for linear processes. J. Time Ser. Anal.30 (2009) 534–558. Zbl1224.62068MR2560417
  32. [32] A. Scherrer, Analyses statistiques des communications sur puce. Ph.D. thesis, École normale supérieure de Lyon (2006). Available on http://www.ens-lyon.fr/LIP/Pub/Rapports/PhD/PhD2006/PhD2006-09.pdf. 
  33. [33] M.S. Taqqu, A representation for self-similar processes. Stoch. Proc. Appl.7 (1978) 55–64. Zbl0373.60048MR492691
  34. [34] M.S. Taqqu, Central limit theorems and other limit theorems for functionals of gaussian processes. Z. Wahrsch. verw. Gebiete 70 (1979) 191–212. Zbl0575.60024
  35. [35] G.W. Wornell and A.V. Oppenheim, Estimation of fractal signals from noisy measurements using wavelets. IEEE Trans. Signal Process.40 (1992) 611–623. 

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