Hurst index estimation for self-similar processes with long-memory

Recent Development In Stochastic Dynamics And Stochastic Analysis

Volumn , Issue , 2010, Pages 91-117

  Chronopoulou, Alexandra ^a   Viens, Frederi G ^a  

^a PURDUE UNIVERSITY   (United States)

Author keywords

Fractional Brownian motion; Hermite process; Hurst parameter; Long memory; Malliavin calculus; Multiple stochastic integral; Non central limit theorem; Parameter estimation; Quadratic variation; Self similar process

Indexed keywords

EID: 85123566150     PISSN: None     EISSN: None     Source Type: Book    
DOI: None     Document Type: Chapter

References (29)

1. J-M. Bardet and C.A. Tudor (2008): A wavelet analysis of the Rosenblatt process: chaos expansion and estimation of the self-similarity parameter.Preprint.
2. J.B. Bassingthwaighte and G.M. Raymond (1994): Evaluating rescaled range analysis for time series, Annals of Biomedical Engineering, 22, 432-444.
3. J. Beran (1994): Statistics for Long-Memory Processes. Chapman and Hall.
4. J.-C. Breton and I. Nourdin (2008): Error bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion. Electronic Communications in Probability, 13, 482-493.
5. P. Breuer and P. Major (1983): Central limit theorems for nonlinear functionals of Gaussian fields. J. Multivariate Analysis, 13 (3), 425-441.
6. A. Chronopoulou, C.A. Tudor and F. Viens (2009): Application of Malliavin calculus to long-memory parameter estimation for non-Gaussian processes. Comptes rendus -Mathematique 347, 663-666.
7. A. Chronopoulou, C.A. Tudor and F. Viens (2009): Variations and Hurst index estimation for a Rosenblatt process using longer filters.Preprint.
8. J.F. Coeurjolly (2001): Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Statistical Inference for Stochastic Processes, 4, 199-227.
9. R. Dahlhaus (1989): Efficient parameter estimation for self-similar processes. Annals of Statistics, 17, 1749-1766.
10. R.L. Dobrushin and P. Major (1979): Non-central limit theorems for non-linear functionals of Gaussian fields. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 50, 27-52.
11. P. Flandrin (1993): Fractional Brownian motion and wavelets. Wavelets, Fractals and Fourier transforms. Clarendon Press, Oxford, 109-122.
12. R. Fox, M. Taqqu (1985): Non-central limit theorems for quadratic forms in random variables having long-range dependence. Probab. Th. Rel. Fields, 13, 428-446.
13. Hurst, H. (1951): Long Term Storage Capacity of Reservoirs, Transactions of the American Society of Civil Engineers, 116, 770-799.
14. A.K. Louis, P. Maass, A. Rieder (1997): Wavelets: Theory and applications Pure & Applied Mathematics. Wiley-Interscience series of texts, monographs & tracts.
15. M. Maejima and C.A. Tudor (2007): Wiener integrals and a Non-Central Limit Theorem for Hermite processes, Stochastic Analysis and Applications, 25 (5), 1043-1056.
16. B.B. Mandelbrot (1975): Limit theorems of the self-normalized range for weakly and strongly dependent processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 31, 271-285.
17. I. Nourdin, D. Nualart and C.A Tudor (2007): Central and Non-Central Limit Theorems for weighted power variations of the fractional Brownian motion. Preprint.
18. I. Nourdin, G. Peccati and A. Réveillac (2008): Multivariate normal approximation using Stein's method and Malliavin calculus. Ann. Inst. H. Poincaré Probab. Statist., 18 pages, to appear.
19. D. Nualart (2006): Malliavin Calculus and Related Topics. Second Edition. Springer.
20. D. Nualart and G. Peccati (2005): Central limit theorems for sequences of multiple stochastic integrals. The Annals of Probability, 33, 173-193.
21. D. Nualart and S. Ortiz-Latorre (2008): Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stochastic Processes and their Applications, 118, 614-628.
22. G. Peccati and C.A. Tudor (2004): Gaussian limits for vector-valued multiple stochastic integrals. Séminaire de Probabilités, XXXIV, 247-262.
23. G. Samorodnitsky and M. Taqqu (1994): Stable Non-Gaussian random variables. Chapman and Hall, London.
24. J. Shao (2007): Mathematical Statistics. Springer.
25. M. Taqqu (1975): Weak convergence to the fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 31, 287-302.
26. C.A. Tudor (2008): Analysis of the Rosenblatt process. ESAIM Probability and Statistics, 12, 230-257.
27. C.A. Tudor and F. Viens (2008): Variations and estimators for the selfsimilarity order through Malliavin calculus. Annals of Probability, 34 pages, to appear.
28. C.A. Tudor and F. Viens (2008): Variations of the fractional Brownian motion via Malliavin calculus. Australian Journal of Mathematics, 13 pages, to appear.
29. P. Whittle (1953): Estimation and information in stationary time series. Ark. Mat., 2, 423-434.