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

TL;DR

This paper introduces a new, robust method based on max self-similarity for estimating the tail index of heavy-tailed, dependent time series data, with proven consistency and asymptotic properties.

Contribution

The paper presents a novel max-spectrum estimator for heavy-tail exponents that is consistent, asymptotically normal, and applicable to a broad class of dependent time series.

Findings

Estimator is consistent for m-dependent series.

Estimator is asymptotically normal.

Effective on synthetic and real data.

Abstract

This paper addresses the problem of estimating the tail index 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.