Long‐range Dependence: Revisiting Aggregation with Wavelets

Abstract

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.