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

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.