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A new distribution-based test of self-similarity


  • 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.

Suggested Citation

  • Bianchi, Sergio, 2004. "A new distribution-based test of self-similarity," MPRA Paper 16640, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:16640

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    References listed on IDEAS

    1. Lobato, I. & Robinson, P. M., 1996. "Averaged periodogram estimation of long memory," Journal of Econometrics, Elsevier, vol. 73(1), pages 303-324, July.
    2. 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.
    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. U. A. Muller & M. M. Dacorogna & R. D. Dave & O. V. Pictet & R. B. Olsen & J.R. Ward, "undated". "Fractals and Intrinsic Time - a Challenge to Econometricians," Working Papers 1993-08-16, Olsen and Associates.
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    Cited by:

    1. Myoungji Lee & Marc G. Genton & Mikyoung Jun, 2016. "Testing Self-Similarity Through Lamperti Transformations," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 21(3), pages 426-447, September.
    2. Roy Cerqueti & Giulia Rotundo, 2015. "A review of aggregation techniques for agent-based models: understanding the presence of long-term memory," Quality & Quantity: International Journal of Methodology, Springer, vol. 49(4), pages 1693-1717, July.

    More about this item


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

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes


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