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Hurst exponent estimation of locally self-similar Gaussian processes using sample quantiles. (English) Zbl 1157.60034
Author’s abstract: This paper is devoted to the introduction of a new class of consistent estimators of the fractal dimension of locally self-similar Gaussian processes. These estimators are based on convex combinations of sample quantiles of discrete variations of a sample path over a discrete grid of the interval $$[0, 1]$$. We derive the almost sure convergence and the asymptotic normality for these estimators. The key-ingredient is a Bahadur representation for sample quantiles of nonlinear functions of Gaussian sequences with correlation function decreasing as $$k ^{- \alpha} L(k)$$ for some $$\alpha >0$$ and some slowly varying function $$L(\cdot )$$.

##### MSC:
 60G18 Self-similar stochastic processes 62G30 Order statistics; empirical distribution functions
SimEstFBM; LASS
Full Text:
##### References:
 [1] Alòs, E., Mazet, O. and Nualart, D. (1999). Stochastic calculus with respect to fractional Brownian motion with Hurst parameter lesser than \frac{1}{2} . Stoch. Proc. Appl. 86 121-139. · Zbl 1028.60047 [2] Antoniadis, A., Berruyer, J. and Carmona, R. (1992). Régression non linéaire et applications . Editions Economica, Paris. [3] Arcones, M. A. (1994). Limit theorems for nonlinear functionals of stationary Gaussian field of vectors. Ann. Probab. 22 2242-2274. · Zbl 0839.60024 [4] Bahadur, R. R. (1966). A note on quantiles in large samples. Ann. Math. Statist. 37 577-580. · Zbl 0147.18805 [5] Bardet, J.-M., Lang, G., Oppenheim, G., Philippe, A., Stoev, S. and Taqqu, M. (2003). Semi-parametric estimation of the long-range dependence parameter: A survey. In Theory and Applications of Long-Range Dependence (P. Doukhan, G. Oppenheim and M. S. Taqqu, eds.) 557-577. Birkhäuser, Boston. · Zbl 1032.62077 [6] Beran, J. (1994). Statistics for Long Memory Processes . Chapman and Hall, London. · Zbl 0869.60045 [7] Breuer, P. and Major, P. (1983). Central limit theorems for non-linear functionals of Gaussian fields. J. Multivariate Anal. 13 425-441. · Zbl 0518.60023 [8] Coeurjolly, J.-F. (2000). Simulation and identification of the fractional Brownian motion: A bibliographical and comparative study. J. Statist. Soft. 5 1-53. [9] Coeurjolly, J.-F. (2001). Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Statist. Inference Stoch. Process. 4 199-227. · Zbl 0984.62058 [10] Coeurjolly, J.-F. (2007). Hurst exponent estimation of locally self-similar Gaussian processes using sample quantiles. Available at http://fr.arxiv.org/abs/math.ST/0506290. · Zbl 1157.60034 [11] Constantine, A. G. and Hall, P. (1994). Characterizing surface smoothness via estimation of effective fractal dimension. J. Roy. Statist. Soc. Ser. B 56 97-113. JSTOR: · Zbl 0804.62079 [12] Dacunha-Castelle, D. and Duflo, M. (1982). Exercices de probabilités et statistiques. Tome 1. Problèmes à temps fixe . Collection Mathémathiques Appliquées pour la Maîtrise, Masson, Paris. · Zbl 0494.62001 [13] Daubechies, I. (1998). Ten Lectures on Wavelets . SIAM, Philadelphia. · Zbl 0776.42018 [14] Flandrin, P. (1992). Wavelet analysis of fractional Brownian motion. IEE Trans. Inform. Theory 38 910-917. · Zbl 0743.60078 [15] Geweke, J. and Porter-Hudak, S. (1983). The estimation and application of long-memory time-series models. J. Time Ser. Anal. 4 221-238. · Zbl 0534.62062 [16] Hesse, C. H. (1990). A Bahadur-type representation for empirical quantiles of a large class of stationary, possibly infinite-variance, linear processes. Ann. Statist. 18 1188-1202. · Zbl 0712.62042 [17] Ho, H. C. and Hsing, T. (1996). On the asymptotic expansion of the empirical process of long-memory moving averages. Ann. Statist. 24 992-1024. · Zbl 0862.60026 [18] Istas, J. and Lang, G. (1997). Quadratic variations and estimation of the Hölder index of a Gaussian process. Ann. Inst. H. Poincaré Probab. Statist. 33 407-436. · Zbl 0882.60032 [19] Kent, J. T. and Wood, A. T. A. (1997). Estimating the fractal dimension of a locally self-similar Gaussian process using increments. J. Roy. Statist. Soc. Ser. B 59 679-700. JSTOR: · Zbl 0889.62072 [20] Kiefer, J. (1967). On Bahadur’s representation of sample quantiles. Ann. Math. Statist. 38 1323-1342. · Zbl 0158.37005 [21] Lim, S. C. (2001). Fractional Brownian motion and multifractional of Riemann-Liouville type. J. Phys. A: Math. Gen. 34 1301-1310. · Zbl 0965.82009 [22] Mandelbrot, B. and Van Ness, J. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 422-437. JSTOR: · Zbl 0179.47801 [23] Sen, P. K. and Ghosh, M. (1971). On bounded length sequential confidence intervals based on one-sample rank order statistics. Ann. Math. Statist. 42 189-203. · Zbl 0223.62100 [24] Sen, P. K. (1972). On the Bahadur representation of sample quantiles for sequences of \varphi -mixing random variables. J. Multivariate Anal. 2 77-95. · Zbl 0226.60050 [25] Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics . Wiley, New York. · Zbl 0538.62002 [26] Stoev, S., Taqqu, M., Park, C., Michailidis, G. and Marron, J. S. (2006). LASS: A tool for the local analysis of self-similarity. Comput. Statist. Data Anal. 50 2447-2471. · Zbl 1445.62240 [27] Taqqu, M. S. (1977). Law of the iterated logarithm for sums of non-linear functions of Gaussian variables that exhibit a long range dependence. Z. Wahrsch. Verw. Gebiete 40 203-238. · Zbl 0358.60048 [28] Wood, A. T. A. and Chan, G. (1994). Simulation of stationary Gaussian processes in [0, 1] d . J. Comput. Graph. Statist. 3 409-432. JSTOR: [29] Yoshihara, K.-I. (1995). The Bahadur representation of sample quantiles for sequences of strongly mixing random variables. Stat. Probab. Lett. 24 299-304. · Zbl 0835.62048 [30] Wu, W.-B. (2005). On the Bahadur representation of sample quantiles for dependent sequences. Ann. Statist. 33 1934-1963. · Zbl 1080.62024
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