Local estimation of the Hurst index of multifractional brownian motion by increment ratio statistic method
Pierre Raphaël Bertrand; Mehdi Fhima; Arnaud Guillin
ESAIM: Probability and Statistics (2013)
- Volume: 17, page 307-327
- ISSN: 1292-8100
Access Full Article
topAbstract
topHow to cite
topBertrand, Pierre Raphaël, Fhima, Mehdi, and Guillin, Arnaud. "Local estimation of the Hurst index of multifractional brownian motion by increment ratio statistic method." ESAIM: Probability and Statistics 17 (2013): 307-327. <http://eudml.org/doc/274365>.
@article{Bertrand2013,
abstract = {We investigate here the central limit theorem of the increment ratio statistic of a multifractional Brownian motion, leading to a CLT for the time varying Hurst index. The proofs are quite simple relying on Breuer–Major theorems and an original freezing of time strategy. A simulation study shows the goodness of fit of this estimator.},
author = {Bertrand, Pierre Raphaël, Fhima, Mehdi, Guillin, Arnaud},
journal = {ESAIM: Probability and Statistics},
keywords = {increment ratio statistic; fractional brownian motion; local estimation; multifractional brownian motion; wavelet series representation; multifractional Brownian motion},
language = {eng},
pages = {307-327},
publisher = {EDP-Sciences},
title = {Local estimation of the Hurst index of multifractional brownian motion by increment ratio statistic method},
url = {http://eudml.org/doc/274365},
volume = {17},
year = {2013},
}
TY - JOUR
AU - Bertrand, Pierre Raphaël
AU - Fhima, Mehdi
AU - Guillin, Arnaud
TI - Local estimation of the Hurst index of multifractional brownian motion by increment ratio statistic method
JO - ESAIM: Probability and Statistics
PY - 2013
PB - EDP-Sciences
VL - 17
SP - 307
EP - 327
AB - We investigate here the central limit theorem of the increment ratio statistic of a multifractional Brownian motion, leading to a CLT for the time varying Hurst index. The proofs are quite simple relying on Breuer–Major theorems and an original freezing of time strategy. A simulation study shows the goodness of fit of this estimator.
LA - eng
KW - increment ratio statistic; fractional brownian motion; local estimation; multifractional brownian motion; wavelet series representation; multifractional Brownian motion
UR - http://eudml.org/doc/274365
ER -
References
top- [1] P. Abry, P. Flandrin, M.S. Taqqu and D. Veitch, Self-similarity and long-range dependence through the wavelet lens, in Theory and applications of long-range dependenc. Birkhauser, Boston (2003). Zbl1029.60028MR1957507
- [2] M.A. Arcones, Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors. Ann. Probab.22 (1994) 2242–2274. Zbl0839.60024MR1331224
- [3] A. Ayache and M.S. Taqqu, Rate optimality of wavelet series approximations of fractional Brownian motions. J. Fourier Anal. Appl.9 (2003) 451–471. Zbl1050.60043MR2027888
- [4] A. Ayache and M.S. Taqqu, Multifractional process with random exponent. Publ. Math.49 (2005) 459–486. Zbl1082.60032MR2177638
- [5] A. Ayache, P. Bertrand and J. Lévy-Véhel, A central limit theorem for the generalized quadratic variation of the step fractional Brownian motion. Stat. Inference Stoch. Process.10 (2007) 1–27. Zbl1115.60024MR2269602
- [6] J.M. Bardet and P.R. Bertrand, Definition, properties and wavelet analysis of multiscale fractional Brownian motions. Fractals15 (2007) 73–87. Zbl1142.60329MR2281946
- [7] J.M. Bardet and P.R. Bertrand, Identification of the multiscale fractional Brownian motion with biomechanical applications. J. Time Ser. Anal.28 (2007) 1–52. Zbl1164.62034MR2332850
- [8] J.M. Bardet and P.R. Bertrand, A nonparametric estimator of the spectral density of a continuous-time Gaussian process observed at random times. Scand. J. Stat.37 (2010) 458–476. Zbl1226.60027MR2724508
- [9] J.M. Bardet and D. Surgailis, Nonparametric estimation of the local hurst function of multifractional Gaussian processes, Stoch. Proc. Appl.123 (2013) 1004–1045. Zbl1257.62034MR3005013
- [10] J.M. Bardet and D. Surgailis, Measuring roughness of random paths by increment ratios. Bernoulli17 (2011) 749–780. Zbl1248.60042MR2787614
- [11] A. Bégyn, Functional limit theorems for generalized quadratic variations of Gaussian processes. Stoch. Proc. Appl.117 (2007) 1848–1869. Zbl1129.60032MR2437732
- [12] A. Benassi, S. Jaffard and D. Roux, Gaussian processes and pseudodifferential elliptic operators. Rev. Mat. Iberoam.13 (1997) 19–81. Zbl0880.60053MR1462329
- [13] A. Benassi, S. Cohen and J. Istas, Identifying the multifractional function of a Gaussian process. Stat. Probab. Lett.39 (1998) 337–345. Zbl0931.60022MR1646220
- [14] P.R. Bertrand, A. Hamdouni and S. Khadhraoui, Modelling NASDAQ series by sparse multifractional Brownian motion. Method. Comput. Appl. Probab.14 (2012) 107–124. Zbl1241.62143MR2875093
- [15] H. Biermé, A. Bonami and J. Leon, Central limit theorems and quadratic variations in terms of spectral density. Electronic Journal of Probability16 (2011) 362–395. Zbl1238.60027MR2774094
- [16] Pa. Billingsley, Probability and measure, 2nd edition. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons Inc., New York (1986). Zbl0649.60001MR830424
- [17] K. Bružaitė and M. Vaičiulis, The increment ratio statistic under deterministic trends. Lith. Math. J.48 (2008) 256–269. Zbl05834354
- [18] G. Chan and A.T.A. Wood, Simulation of multifractal Brownian motions, Proc. of Computational Statistics (1998) 233–238. Zbl0952.65006
- [19] P. Cheridito, Arbitrage in fractional Brownian motion models. Finance Stoch.7 (2003) 533–553. Zbl1035.60036MR2014249
- [20] J.F. Coeurjolly, Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Stat. Inference Stoch. Process.4 (2001) 199–227. Zbl0984.62058MR1856174
- [21] J.-F. Coeurjolly, Identification of multifractional Brownian motions. Bernoulli11 (2005) 987–1008. Zbl1098.62109MR2188838
- [22] S. Cohen, From self-similarity to local self-similarity: the estimation problem, Fractal: Theory and Applications in Engineering, edited by M. Dekking, J. Lévy Véhel, E. Lutton and C. Tricot. Springer Verlag (1999). Zbl0965.60073MR1726364
- [23] H. Cramèr and M.R. Leadbetter, Stationary and Related Stochastic Processes. Sample Function Properties and Their Applications, Wiley and Sons, London (1967). Zbl0162.21102MR217860
- [24] M. Fhima, Ph.D. thesis (2011) in preparation.
- [25] X. Guyon and J. Leon, Convergence en loi des h-variations d’un processus Gaussien stationnaire. Ann. Inst. Henri Poincaré25 (1989) 265–282. Zbl0691.60017MR1023952
- [26] J. Istas and G. Lang, Quadratic variations and estimation of the hölder index of a Gaussian process. Ann. Inst. Henri Poincaré33 (1997) 407–436. Zbl0882.60032MR1465796
- [27] A.N. Kolmogorov, Wienersche spiralen und einige andere interessante kurven im hilbertschen raum. C.R. (Doklady) Acad. URSS (N.S.) 26 (1940) 115–118. Zbl66.0552.03MR3441JFM66.0552.03
- [28] J. Lévy-Véhel and R.F. Peltier, Multifractional Brownian motion: definition and preliminary results. Techn. Report RR-2645, INRIA (1996).
- [29] B. Mandelbrot and J. Van Ness, Fractional Brownian motions, fractional noises and applications. SIAM Review10 (1968) 422–437. Zbl0179.47801MR242239
- [30] Y. Meyer, F. Sellan and M.S. Taqqu, Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motions. J. Fourier Anal. Appl.5 (1999) 465–494. Zbl0948.60026MR1755100
- [31] I. Nourdin and G. Peccati, Stein’s method on wiener chaos. Probab. Theory Relat. Fields145 (2009) 75–118. Zbl1175.60053MR2520122
- [32] I. Nourdin, G. Peccati and M. Podolskij, Quantitative Breuer-Major theorems, HAL: hal-00484096, version 2 (2010). Zbl1225.60045MR2770907
- [33] G. Peccati and C.A. Tudor, Gaussian limits for vector-valued multiple stochastic integrals. Séminaire de Probabilités XXXVIII, Lecture Notes Math. 1857 (2005) 247–262. Zbl1063.60027MR2126978
- [34] G. Samorodnitsky and M.S. Taqqu, Stable non-Gaussian random processes. Chapman & Hall (1994). Zbl0925.60027MR1280932
- [35] A.S. Stoev and M.S. Taqqu, How rich is the class of multifractional brownian motions. Stoch. Proc. Appl.116 (2006) 200–221. Zbl1094.60024MR2197974
- [36] M. Stoncelis and M. Vaičiulis, Numerical approximation of some infinite Gaussian series and integrals. Nonlinear Anal.: Modelling and Control 13 (2008) 397–415. Zbl1177.62113MR2457610
- [37] D. Surgailis, G. Teyssière and M. Vaičiulis, The increment ratio statistic. J. Multivar. Anal.99 (2008) 510–541. Zbl1132.62074MR2396977
- [38] A.M. Yaglom, Some classes of random fields in n-dimensional space, related to stationary random processes. Theory Probab. Appl.2 (1957) 273–320.
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.