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

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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 ; Diffusion Processes 
Item ID:  16640 
Depositing User:  Sergio Bianchi 
Date Deposited:  11. Aug 2009 15:10 
Last Modified:  16. Feb 2013 05:23 
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URI:  https://mpra.ub.unimuenchen.de/id/eprint/16640 