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A wavelet-based joint estimator of the parameters of long-range dependence. (English) Zbl 0945.94006
Summary: 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 recently proposed wavelet-based estimator of the scaling parameter [P. Abry and D. Veitch, IEEE Trans. Inf. Theory 44, 2-15 (1998; Zbl 0905.94006)] 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 $\left(O\left(n\right)\right)$ 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.

MSC:
 94A12 Signal theory (characterization, reconstruction, filtering, etc.) 62M09 Non-Markovian processes: estimation 62M10 Time series, auto-correlation, regression, etc. (statistics)