On the Estimation of the Heavy–Tail Exponent in Time Series using the Max–Spectrum

This paper addresses the problem of estimating the tail index α of distributions with heavy, Pareto–type tails for dependent data, that is of interest in the areas of finance, insurance, environmental monitoring and teletraffic analysis. A novel approach based on the max self–similarity scaling behavior of block maxima is introduced. The method exploits the increasing lack of dependence of maxima over large size blocks, which proves useful for time series data. We establish the consistency of the proposed max–spectrum estimator for certain classes of dependent time series and demonstrate its robustness to short–lived contaminations in the data. The max–spectrum estimator exhibits linear computational time and memory complexity and can be calculated in a sequential manner, that makes it particularly well suited both for massive, as well as streaming data sets. Further, it provides a natural time–scale perspective in the analysis of heavy–tailed time series, not available in Hill–type techniques. The performance of the max–spectrum estimator is illustrated on synthetic and real data.