Divergence based Robust Estimation of the Tail Index through An Exponential Regression Model

TL;DR

This paper introduces a new robust estimator for the tail index in extreme value theory that performs well across all tail types, reducing bias and increasing robustness compared to existing methods.

Contribution

It proposes a novel, robust, and bias-reduced tail index estimator applicable to all three tail types using an exponential regression model with density power divergence.

Findings

The estimator is robust against outliers across all tail types.

Simulation studies show reduced bias compared to existing estimators.

Application to real data demonstrates practical effectiveness.

Abstract

The extreme value theory is very popular in applied sciences including Finance, economics, hydrology and many other disciplines. In univariate extreme value theory, we model the data by a suitable distribution from the general max-domain of attraction (MAD) characterized by its tail index; there are three broad classes of tails -- the Pareto type, the Weibull type and the Gumbel type. The simplest and most common estimator of the tail index is the Hill estimator that works only for Pareto type tails and has a high bias; it is also highly non-robust in presence of outliers with respect to the assumed model. There have been some recent attempts to produce asymptotically unbiased or robust alternative to the Hill estimator; however all the robust alternatives work for any one type of tail. This paper proposes a new general estimator of the tail index that is both robust and has smaller bias under all the three tail types compared to the existing robust estimators. This essentially produces a robust generalization of the estimator proposed by Matthys and Beirlant (2003) under the same model approximation through a suitable exponential regression framework using the density power divergence. The robustness properties of the estimator are derived in the paper along with an extensive simulation study. A method for bias correction is also proposed with application to some real data examples.