Fast and unbiased estimator of the time-dependent Hurst exponent. (English) Zbl 1452.62111
Summary: We combine two existing estimators of the local Hurst exponent to improve both the goodness of fit and the computational speed of the algorithm. An application with simulated time series is implemented, and a Monte Carlo simulation is performed to provide evidence of the improvement.{
©2018 American Institute of Physics}
©2018 American Institute of Physics}
MSC:
| 62-08 | Computational methods for problems pertaining to statistics |
| 60G22 | Fractional processes, including fractional Brownian motion |
| 62M09 | Non-Markovian processes: estimation |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
References:
| [1] | Péltier, R.; Lévy Véhel, J., A New Method for Estimating the Parameter of Fractional Brownian Motion, 1-27, (1994), INRIA |
| [2] | Istas, J.; Lang, G., Variations quadratiques et estimation de l’exposant de Hölder local d’un processus Gaussien, Ann. Inst. Henri Poincaré, 33, 407-436, (1997) · Zbl 0882.60032 · doi:10.1016/S0246-0203(97)80099-4 |
| [3] | Benassi, A.; Jaffard, S.; Roux, D., Elliptic gaussian random processes, Rev. Math. Iberoam., 13, 1, 19-89, (1997) · Zbl 0880.60053 · doi:10.4171/RMI/217 |
| [4] | Coeurjolly, J. F., Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths, Stat. Inference Stochastic Processes, 4, 199-227, (2001) · Zbl 0984.62058 · doi:10.1023/A:1017507306245 |
| [5] | Gloter, A.; Hoffmann, M., Estimation of the Hurst parameter from discrete noisy data, Ann. Stat., 35, 1947-1974, (2007) · Zbl 1126.62073 · doi:10.1214/009053607000000316 |
| [6] | Coeurjolly, J. F., Hurst exponent estimation of locally self-similar gaussian processes using sample quantiles, Ann. Stat., 36, 1404-1434, (2008) · Zbl 1157.60034 · doi:10.1214/009053607000000587 |
| [7] | Péltier, R.; Lévy Véhel, J., Multifractional Brownian Motion: Definition and Preliminary Results, 1-39, (1995), INRIA |
| [8] | Muzy, J. F.; Bacry, E.; Arneodo, A., Wavelets and multifractal formalism for singular signals: Application to turbulence data, Phys. Rev. Lett., 67, 3515-3518, (1991) · doi:10.1103/PhysRevLett.67.3515 |
| [9] | Arneodo, A.; Argoul, F.; Bacry, E.; Elezgaray, J.; Muzy, J.-F.; Diderot, P., Ondelettes, Multifractales et Turbulence. De lADN auz Croissances Cristallines, (1995), Editeurs des Sciences et des Arts |
| [10] | Abry, P.; Diderot, P., Ondelettes et TurbulencesMultirésolutions, Algorithmes de Décompositions, Invariance déchelle et Signauz de Pression, (1997), Editeurs des Sciences et des Arts · Zbl 0917.65115 |
| [11] | Lee, K., Characterization of turbulence stability through the identification of multifractional Brownian motions, Nonlinear Processes Geophys., 20, 97-106, (2013) · doi:10.5194/npg-20-97-2013 |
| [12] | Véhel, J. L.; Fisher, Y., Introduction to the multifractal analysis of images, Fractal Image Encoding and Analysis, (1998), Springer-Verlag: Springer-Verlag, New York |
| [13] | Tan, Z.; Atto, A. M.; Alata, O.; Moreaud, M., Arfbf model for non stationary random fields and application in HRTEM images, 2651-2655, (2015) |
| [14] | Combrexelle, S.; Wendt, H.; Dobigeon, N.; Tourneret, J.-Y.; McLaughlin, S.; Abry, P., Bayesian estimation of the multifractality parameter for image texture using a whittle approximation, IEEE Trans. Image Process., 24, 2540-2551, (2015) · Zbl 1408.94112 · doi:10.1109/TIP.2015.2426021 |
| [15] | Reza, S. M. S.; Islam, A.; Iftekharuddin, K. M.; Di Ieva, A., Texture estimation for abnormal tissue segmentation in brain mri, The Fractal Geometry of the Brain, 333-349, (2016), Springer New York: Springer New York, New York, NY |
| [16] | Shboul, Z. A.; Reza, S. M. S.; Iftekharuddin, K. M.; Tavares, J. M. R.; Natal Jorge, R., Quantitative mr image analysis for brain tumor, VipIMAGE 2017, 10-18, (2018), Springer International Publishing: Springer International Publishing, Cham |
| [17] | Lopes, R.; Betrouni, N., Fractal and multifractal analysis: A review, Med. Image Anal., 13, 634-649, (2009) · doi:10.1016/j.media.2009.05.003 |
| [18] | Bender, V. E.; Sottinen T, C.; Nunno, G. Di; Øksendal, B., Fractional processes as models in stochastic finance, Advanced Mathematical Methods for Finance, (2011), Springer, Berlin/Heidelberg · Zbl 1239.91001 |
| [19] | Bianchi, S.; Pianese, A., Multifractional processes in finance, Risk Decis. Anal., 5, 1-22, (2014) · Zbl 1409.91294 · doi:10.3233/RDA-130097 |
| [20] | Tapiero, C. S.; Tapiero, O. J.; Jumarie, G., The price of granularity and fractional finance, Risk Decis. Anal., 6, 7-21, (2016) · doi:10.3233/RDA-150112 |
| [21] | Lux, T.; Segnon, M.; Chen, S.-H.; Kaboudan, M.; Du, Y.-R., Multifractal models in finance: Their origin, properties and applications, The Oxford Handbook of Computational Economics and Finance, (2018), Oxford Handbooks |
| [22] | Benassi, A.; Cohen, S.; Istas, J., Identifying the multifractional function of a Gaussian process, Stat. Probab. Lett., 39, 337-345, (1998) · Zbl 0931.60022 · doi:10.1016/S0167-7152(98)00078-9 |
| [23] | Benassi, A.; Bertrand, P.; Cohen, S.; Istas, J., “Identification Hurst index a step fractional Brownian motion,” Stat. Inference Stochastic Processes, 3, 101-111, (2000) · Zbl 0982.60081 · doi:10.1023/A:1009997729317 |
| [24] | Coeurjolly, J. F., Identification of multifractional Brownian motion, Bernoulli, 11, 987-1008, (2005) · Zbl 1098.62109 · doi:10.3150/bj/1137421637 |
| [25] | Bianchi, S., Pathwise identification of the memory function of the multifractional Brownian motion with application to finance, Int. J. Theor. Appl. Finance, 8, 255-281, (2005) · Zbl 1100.91037 · doi:10.1142/S0219024905002937 |
| [26] | Bianchi, S.; Pantanella, A.; Pianese, A., Modeling stock prices by multifractional Brownian motion: An improved estimation of the pointwise regularity, Quant. Finance, 13, 1317-1330, (2013) · Zbl 1281.91083 · doi:10.1080/14697688.2011.594080 |
| [27] | Ayache, A.; Lévy Véhel, J., On the identification of the pointwise Hölder exponent of the generalized multifractional Brownian motion, Stochastic Processes Appl., 111, 119-156, (2004) · Zbl 1079.60029 · doi:10.1016/j.spa.2003.11.002 |
| [28] | Ayache, A.; Benassi, A.; Cohen, S.; Lévy Véhel, J., Regularity and identification of generalized multifractional gaussian processes, Séminaire de Probabilités XXXVIII. Lecture Notes in Mathematics, 1857, 290-312, (2005), Springer, Berlin/Heidelberg · Zbl 1076.60026 |
| [29] | Ayache, A.; Bertrand, P.; Lévy Véhel, J., A central limit theorem for the quadratic variations of the step fractional Brownian motion, Stat. Inference Stochastic Processes, 10, 1-27, (2007) · Zbl 1115.60024 · doi:10.1007/s11203-005-0532-2 |
| [30] | Loutridis, S., An algorithm for the characterization of time-series based on local regularity, Phys. A: Stat. Mech. Appl., 381, 383-398, (2007) · doi:10.1016/j.physa.2007.03.012 |
| [31] | Garcin, M., Estimation of time-dependent Hurst exponents with variational smoothing and application to forecasting foreign exchange rates, Phys. A: Stat. Mech. its Appl., 483, 462-479, (2017) · Zbl 1499.62118 · doi:10.1016/j.physa.2017.04.122 |
| [32] | Taqqu, M.; Teverovsky, V.; Willinger, W., Estimators for long-range dependence: An empirical study, Fractals, 3, 785-798, (1995) · Zbl 0864.62061 · doi:10.1142/S0218348X95000692 |
| [33] | Rea, W.; Oxley, L.; Reale, M.; Brown, J., Not all estimators are born equal: The empirical properties of some estimators of long memory, Math. Comput. Simul., 93, 29-42, (2013) · Zbl 1499.62320 |
| [34] | Storer, R.; Scansaroli, D.; Dobric, V., New estimators of the Hurst index for fractional Brownian motion, (2014) |
| [35] | Kent, J. T.; Wood, A. T. A., Estimating the fractal dimension of a locally selfsimilar Gaussian process using increments, J. R. Stat. Soc. Ser. B, 59, 679-700, (1997) · Zbl 0889.62072 |
| [36] | Chan, G.; Wood, A. T. A.; Payne, R.; Green, P., Simulation of multifractional Brownian motion, COMPSTAT, 233-238, (1998), Physica-Verlag HD: Physica-Verlag HD, Heidelberg · Zbl 0952.65006 |
| [37] | Bianchi, S.; Pianese, A., Multifractional properties of stock indices decomposed by filtering their pointwise hlder regularity, Int. J. Theor. Appl. Finance, 11, 567-595, (2008) · Zbl 1210.91129 · doi:10.1142/S0219024908004932 |
| [38] | Frezza, M., Goodness of fit assessment for a fractal model of stock markets, Chaos, Solitons Fractals, 66, 41-50, (2014) · Zbl 1349.62578 · doi:10.1016/j.chaos.2014.05.005 |
| [39] | Gerlich, N.; Rostek, S., Estimating serial correlation and self-similarity in financial time seriesa diversification approach with applications to high frequency data, Phys. A: Stat. Mech. Appl., 434, 84-98, (2015) · doi:10.1016/j.physa.2015.03.085 |
| [40] | Bianchi, S.; Pianese, A.; Bensoussan, A.; Guegan, D.; Tapiero, C. S., Asset price modeling: From fractional to multifractional processes, Future Perspectives in Risk Models and Finance, 247-285, (2015), Springer International Publishing: Springer International Publishing, Cham |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
