Bianchi, Sergio (2004): A new distributionbased test of selfsimilarity. Published in: Fractals , Vol. 12, No. 3 (2004): pp. 331346.

PDF
MPRA_paper_16640.pdf Download (1697Kb)  Preview 
Abstract
In studying the scale invariance of an empirical time series a twofold problem arises: it is necessary to test the series for selfsimilarity and, once passed such a test, the goal becomes to estimate the parameter H0 of selfsimilarity. The estimation is therefore correct only if the sequence is truly selfsimilar but in general this is just assumed and not tested in advance. In this paper we suggest a solution for this problem. Given the process {X(t)}, we propose a new test based on the diameter d of the space of the rescaled probability distribution functions of X(t). Two necessary conditions are deduced which contribute to discriminate selfsimilar processes and a closed formula is provided for the diameter of the fractional Brownian motion (fBm). Furthermore, by properly chosing the distance function, we reduce the measure of selfsimilarity to the Smirnov statistics when the onedimensional distributions of X(t) are considered. This permits the application of the wellknown twosided test due to Kolmogorov and Smirnov in order to evaluate the statistical significance of the diameter d, even in the case of strongly dependent sequences. As a consequence, our approach both tests the series for selfsimilarity and provides an estimate of the selfsimilarity parameter.
Item Type:  MPRA Paper 

Original Title:  A new distributionbased test of selfsimilarity 
Language:  English 
Keywords:  Distance, Fractional Brownian motion, KolmogorovSmirnov Test, SelfSimilarity 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C13  Estimation: General C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C14  Semiparametric and Nonparametric Methods: General C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C12  Hypothesis Testing: General C  Mathematical and Quantitative Methods > C2  Single Equation Models; Single Variables > C22  TimeSeries Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models 
Item ID:  16640 
Depositing User:  Sergio Bianchi 
Date Deposited:  11. Aug 2009 15:10 
Last Modified:  16. Feb 2013 05:23 
References:  B. B. Mandelbrot, Une classe de processes stochastiques homothétiques à soi; application à la loi climatologique de H.E. Hurst, Comptes Rendus de l'Académie des Sciences de Paris 260 (1965) 32743277 B. B. Mandelbrot, Sporadic random functions and conditional spectral analysis: selfsimilar examples and limits, Proc. 5th Berkeley Symp. Math. Stat. Prob. 3 (1967) 155179. B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10 (1968) 422437 K. Berkner and C. J. G. Evertsz, Large deviations and selfsimilarity analysis of graphs: DAX stock prices, Chaos Solitons Fractals 6 (1995) 121130 C. J. G. Evertsz, Selfsimilarity of highfrequency USDDEM exchange rates, in Proceedings of the First international Conference on High Frequency Data in Finance (Zurich, Switzerland, 1995) Y. Kumagai, Fractal structure of financial high frequency data, Fractals 10(1) (2002) 1318 U. A. Muller, M. M. Dacorogna, R. B. Olsen, O. V. Pictet, M. Schwarz and C. Morgenegg, Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intradays analysis, J. Bank Fin. 14 (1990) 11891208 U. A. Muller, M. M. Dacorogna, R. D. Dave, R. B. Olsen, O. V. Pictet and J. R. Ward, Fractals and intrinsic time; a challenge to econometricians, in The 34th International Conference of the Applied Econometrics Association, Luxembourg, 1993 A. Razdan, Scaling in the Bombay Stock Exchange Index, Pramana 58(3) (2002) 537544. B. B. Mandelbrot, Gaussian SelfAffinity and Fractals (Springer, New York, 2002). W. Willinger, M. S. Taqqu and A. Erramilli, A bibliographical guide to selfsimilar traffic and performance modeling for highspeed networks, in Stochastic Networks, ads. F. P. Kelly, S. Zachary and I. Ziedins (Oxford University Press, Oxford, 1996), 339366 P. Doukhan, G. Oppenheim and M. S. Taqqu (eds.), LongRange Dependence: Theory and Applications (Birkhauser, 2002) B. B. Mandelbrot, Fractals and Scaling in Finance (Springer, New York, 1997) G. Samorodnitsky and M. S. Taqqu, Stable NonGaussian Random Processes (Chapman & Hall, New York, 1994) P. Embrechts and M. Maejima, Selfsimilar Processes (Princeton University Press, Princeton, 2002) M. S. Taqqu, V. Teverovsky and W. Willinger, Estimators for longrange dependence: an empirical study, Fractals 3(4) (1995) 785798 J. W. Lamperti, Semistable stochastic processes, Trans. Amer. Math. Soc. 104 (1962) 6278 N. V. Smirnov, On the estimation of the discrepancy between empirical curves of distribution for two independent samples, Bull. Math. Univ. Moscow 2 (1939) 314 A. N. Kolmogorov, Foundations of the Theory of Probability (Chelsea Publishing Co., New York, 1933) D. A. Darling, The Kolmogorovsmirnov, CramerVon Mises tests, Ann. Math. Stat. 28 (1957) 823838 A. Wood and G. Chan, Simulation of stationary Gaussian processes in [0, 1]^d, J. Comput. Graph. Stat. 3(4) (1994) 409432 J. Bardet, G. Lang, G. Oppenheim, A. Philippe and M. S. Taqqu, Generators of longrange dependent processes: a survey, in Theory and Application of Long Range Dependence, eds. M. S. Taqqu, G. Oppenheim and P. Doukhan (Birkhauser, Boston, 2002) J. F. Coeurjolly, Simulation and identification of the Fractional Brownian Motion: A Bibliographical and Comparative Study, available at http://www.jstatsoft.org/v05/i07/simest.pdf (2000) R. Jennane, R. Harba and G. Jacquet, Estimation de la qualité des méthodes de synthèses du mouvement Brownien fractionnaire, Traitement du Signal 13(4) (1996) 289302 R. F. Péltier and J. Lévy Véhel, Multfractional Brownian Motion: Definition and Preliminary Results (Rapport de Recherche INRIA n. 2645, 1995) I. Lobato and P. M. Robinson, Averaged periodogram estimation of long memory, J. Econom. 73 (1996) 303324 J. Beran, Statistics for Long Memory Processes (Chapman and Hall, London, 1994) A. Feuerverger, P. Hall and T. A. Wood, Estimation of fractal index and fractal dimension of a Gaussian process by counting the number of level crossings, J. Times Ser. Anal. 15(6) (1994) 587606 J. F. Coeurjolly, Identification du Mouvement Brownien Fractionnaire par Variations discrete (Rapport de Recherche 1016M, IMAG, Grenoble, (1999) I. P. Cornfeld, Y. G. Sinai and S. V. Fomin, Ergodic Theory (Springer Verlag, Berlin, 1982) W. Feller, An Introduction to Probability Theory and its Applications, 2nd edn. (John Wiley & Sons, New York, 1971) B. V. Gnedenko and V. C. Koroljuk, On the maximal deviation between two empirical distributions, Doklady Akad. Nauk SSSR 80 (1951) 525528 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/16640 