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 and asymptotic normality of the proposed max‐spectrum estimator for a large class of m‐dependent time series, in the regime of intermediate block‐maxima. In the regime of large block‐maxima, we demonstrate the distributional consistency of the estimator for a broad range of time series models including linear processes. The max‐spectrum estimator is a robust and computationally efficient tool, which provides a novel time‐scale perspective to the estimation of the tail exponents. Its performance is illustrated over synthetic and real data sets. Copyright © 2009 John Wiley & Sons, Ltd.