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Long-range dependence: revisiting aggregation with wavelets. (English) Zbl 0910.62080

Summary: The aggregation procedure is a natural way to analyse signals which exhibit long-range-dependent features and has been used as a basis for estimation of the Hurst parameter, $H$. In this paper it is shown how aggregation can be naturally rephrased within the wavelet transform framework, being directly related to approximations of the signal in the sense of a Haar multiresolution analysis.

A natural wavelet-based generalization to traditional aggregation is then proposed: ‘a-aggregation’. It is shown that a-aggregation cannot lead to good estimators of $H$, and so a new kind of aggregation, ‘d-aggregation’, is defined, which is related to the details rather than the approximations of a multiresolution analysis. An estimator of $H$ based on d-aggregation has excellent statistical and computational properties, whilst preserving the spirit of aggregation. The estimator is applied to telecommunications network data.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. (statistics) 42C40 Wavelets and other special systems