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Wavelet-based analysis of non-Gaussian long-range dependent processes and estimation of the hurst parameter

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Abstract

In this contribution, the statistical performance of the wavelet-based estimation procedure for the Hurst parameter is studied for non-Gaussian long-range dependent processes obtained from point transformations of Gaussian processes. The statistical properties of the wavelet coefficients and the estimation performance are compared both for processes having the same covariance but different marginal distributions and for processes having the same covariance and same marginal distributions but obtained from different point transformations, analyzed using mother wavelets with different number of vanishing moments. It is shown that the reduction of the dependence range from long to short by increasing the number of vanishing moments, observed for Gaussian processes, and at the origin of the popularity of the wavelet-based estimator, does not hold in general for non-Gaussian processes. Crucially, it is also observed that the Hermite rank of the point transformation impacts significantly the statistical properties of the wavelet coefficients and the estimation performance and also that processes having identical marginal distributions and covariance function can yet yield significantly different estimation performance. These results are interpreted in the light of central and noncentral limit theorems that are fundamental when dealing with long-range dependent processes. Moreover, it will be shown that, on condition that estimation is performed using a range of scales restricted to the coarsest practically available, an approximate, yet analytical and simple to use in practice, formula can be proposed for the evaluation of the variance of the wavelet-based estimator of the Hurst parameter.

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Authors and Affiliations

  1. Laboratoire de Physique (CNRS UMR 5672), École Normale Supérieure de Lyon, Lyon, France

    Patrice Abry & Hannes Helgason

  2. Department of Statistics and Operations Research, University of North Carolina, North Carolina, United States

    Vladas Pipiras

Authors
  1. Patrice Abry
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  2. Hannes Helgason
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  3. Vladas Pipiras
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Correspondence to Patrice Abry.

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Work partially supported by the 2007 Young Research Team award granted by Foundation del Duca, Académie des Sciences, Institut de France, and by the NSF grant DMS-060866.

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Abry, P., Helgason, H. & Pipiras, V. Wavelet-based analysis of non-Gaussian long-range dependent processes and estimation of the hurst parameter. Lith Math J 51, 287–302 (2011). https://doi.org/10.1007/s10986-011-9126-4

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