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.Comment: Pre-Print, 35 pages, To appear in "Statistical Methods and
Applications
