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A new distribution-based test of self-similarity


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  • Bianchi, Sergio


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.

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Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 16640.

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Date of creation: 2004
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Publication status: Published in Fractals 3.12(2004): pp. 331-346
Handle: RePEc:pra:mprapa:16640

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

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  1. Akita, Koji & Maejima, Makoto, 2002. "On certain self-decomposable self-similar processes with independent increments," Statistics & Probability Letters, Elsevier, vol. 59(1), pages 53-59, August.
  2. U. A. Muller & M. M. Dacorogna & R. D. Dave & O. V. Pictet & R. B. Olsen & J.R. Ward, . "Fractals and Intrinsic Time - a Challenge to Econometricians," Working Papers 1993-08-16, Olsen and Associates.
  3. Muller, Ulrich A. & Dacorogna, Michel M. & Olsen, Richard B. & Pictet, Olivier V. & Schwarz, Matthias & Morgenegg, Claude, 1990. "Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis," Journal of Banking & Finance, Elsevier, vol. 14(6), pages 1189-1208, December.
  4. Lobato, I. & Robinson, P. M., 1996. "Averaged periodogram estimation of long memory," Journal of Econometrics, Elsevier, Elsevier, vol. 73(1), pages 303-324, July.
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