Divergence based Robust Estimation of Tail Index with Exponential Regression Model

The extreme value theory is becoming very popular in several applied sciences including finance, economy, hydrology etc. In univariate extreme value theory, we model the data by a heavy-tailed distribution characterized by its tail index; there are three broad class of tails – Pareto type, Weibull type and Gumbel type. The simplest and common estimator of the tail index is the Hill estimator that works only for Pareto type tails and has high bias; it is also highly non-robust in presence of the outliers with respect to the model. There are some recent attempts that produces asymptotically unbiased or robust alternative to the hill estimator; however all the robust alternatives works for any one type of tails. This paper proposes a new general estimator of the tail index that is both robust and have less bias under all the three types of tails compared to the existing robust estimator. This essentially produce a robust generalization of the estimator proposed by Matthys and Beirlant (2003) under the same model approximation to a suitable exponential regression framework using the density power divergence. The asymptotic and robustness properties of the estimator are derived in the paper along with an extensive simulation study