Bianchi, Sergio
(2004):
*A new distribution-based test of self-similarity.*
Published in: Fractals
, Vol. 12, No. 3
(2004): pp. 331-346.

Preview |
PDF
MPRA_paper_16640.pdf Download (1MB) | Preview |

## Abstract

In studying the scale invariance of an empirical time series a twofold problem arises: it is necessary to test the series for self-similarity and, once passed such a test, the goal becomes to estimate the parameter H0 of self-similarity. The estimation is therefore correct only if the sequence is truly self-similar 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 self-similar 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 self-similarity to the Smirnov statistics when the one-dimensional distributions of X(t) are considered. This permits the application of the well-known two-sided 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 self-similarity and provides an estimate of the self-similarity parameter.

Item Type: | MPRA Paper |
---|---|

Original Title: | A new distribution-based test of self-similarity |

Language: | English |

Keywords: | Distance, Fractional Brownian motion, Kolmogorov-Smirnov Test, Self-Similarity |

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 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes |

Item ID: | 16640 |

Depositing User: | Sergio Bianchi |

Date Deposited: | 11 Aug 2009 15:10 |

Last Modified: | 26 Sep 2019 20:02 |

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) 3274-3277 B. B. Mandelbrot, Sporadic random functions and conditional spectral analysis: self-similar examples and limits, Proc. 5th Berkeley Symp. Math. Stat. Prob. 3 (1967) 155-179. B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10 (1968) 422-437 K. Berkner and C. J. G. Evertsz, Large deviations and self-similarity analysis of graphs: DAX stock prices, Chaos Solitons Fractals 6 (1995) 121-130 C. J. G. Evertsz, Self-similarity of high-frequency USD-DEM 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) 13-18 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) 1189-1208 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) 537-544. B. B. Mandelbrot, Gaussian Self-Affinity and Fractals (Springer, New York, 2002). W. Willinger, M. S. Taqqu and A. Erramilli, A bibliographical guide to self-similar traffic and performance modeling for high-speed networks, in Stochastic Networks, ads. F. P. Kelly, S. Zachary and I. Ziedins (Oxford University Press, Oxford, 1996), 339-366 P. Doukhan, G. Oppenheim and M. S. Taqqu (eds.), Long-Range 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 Non-Gaussian Random Processes (Chapman & Hall, New York, 1994) P. Embrechts and M. Maejima, Self-similar Processes (Princeton University Press, Princeton, 2002) M. S. Taqqu, V. Teverovsky and W. Willinger, Estimators for long-range dependence: an empirical study, Fractals 3(4) (1995) 785-798 J. W. Lamperti, Semi-stable stochastic processes, Trans. Amer. Math. Soc. 104 (1962) 62-78 N. V. Smirnov, On the estimation of the discrepancy between empirical curves of distribution for two independent samples, Bull. Math. Univ. Moscow 2 (1939) 3-14 A. N. Kolmogorov, Foundations of the Theory of Probability (Chelsea Publishing Co., New York, 1933) D. A. Darling, The Kolmogorov-smirnov, Cramer-Von Mises tests, Ann. Math. Stat. 28 (1957) 823-838 A. Wood and G. Chan, Simulation of stationary Gaussian processes in [0, 1]^d, J. Comput. Graph. Stat. 3(4) (1994) 409-432 J. Bardet, G. Lang, G. Oppenheim, A. Philippe and M. S. Taqqu, Generators of long-range 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) 289-302 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) 303-324 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) 587-606 J. F. Coeurjolly, Identification du Mouvement Brownien Fractionnaire par Variations discrete (Rapport de Recherche 1016-M, 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) 525-528 |

URI: | https://mpra.ub.uni-muenchen.de/id/eprint/16640 |