A wavelet-based tool for the analysis of long-range dependence and a related semi-parametric estimator of the Hurst parameter is introduced. The estimator is shown to be unbiased under very general conditions, and efficient under Gaussian assumptions. It can be implemented very efficiently allowing the direct analysis of very large data sets, and is highly robust against the presence of deterministic trends, as well as allowing their detection and identification. Statistical, computational, and numerical comparisons are made against traditional estimators including that of Whittle. The estimator is used to perform a thorough analysis of the long-range dependence in Ethernet traffic traces. New features are found with important implications for the choice of valid models for performance evaluation. A study of mono versus multifractality is also performed, and a preliminary study of the stationarity with respect to the Hurst parameter and deterministic trends.

A joint estimator is presented for the two parameters that define the long-range dependence phenomenon in the simplest case. The estimator is based on the coefficients of a discrete wavelet decomposition, improving a wavelet-based estimator of the scaling parameter (Abry and Veitch 1998), as well as extending it to include the associated power parameter. An important feature is its conceptual and practical simplicity, consisting essentially in measuring the slope and the intercept of a linear fit after a discrete wavelet transform is performed, a very fast (O(n)) operation. Under well-justified technical idealizations the estimator is shown to be unbiased and of minimum or close to minimum variance for the scale parameter, and asymptotically unbiased and efficient for the second parameter. Through theoretical arguments and numerical simulations it is shown that in practice, even for small data sets, the bias is very small and the variance close to optimal for both parameters. Closed-form expressions are given for the covariance matrix of the estimator as a function of data length, and are shown by simulation to be very accurate even when the technical idealizations are not satisfied. Comparisons are made against two maximum-likelihood estimators. In terms of robustness and computational cost the wavelet estimator is found to be clearly superior and statistically its performance is comparable. We apply the tool to the analysis of Ethernet teletraffic data, completing an earlier study on the scaling parameter alone.

Multifractal analysis is becoming a standard statistical analysis technique. In signal processing, it mostly consists of estimating scaling exponents characterizing scale invariance properties. For practical purposes, confidence intervals in estimation and p values in hypothesis testing are of primary importance. In empirical multifractal analysis, the statistical performance of estimation or test procedures remain beyond analytical derivation because of the theoretically involved nature of multifractal processes. Therefore, the goal of this article is to show how non-parametric bootstrap approaches circumvent such limitations and yield procedures that exhibit satisfactory statistical performance and can hence be practically used on real-life data. Such tools are illustrated at work on the analysis of the multifractal properties of empirical hydrodynamic turbulence data.