In univariate extreme value theory, we model the data by a suitable distribution from the general max-domain of attraction 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 which 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 alternatives to the Hill estimator; however, all the robust alternatives work for any one type of tail. In this talk, we will discuss a general estimator of the tail index that is both robust and has a smaller bias under all three tail types compared to the existing robust estimators. This essentially uses the minimum density power divergence approach and the concept of approximating the extreme value model through a suitable exponential regression framework. The robustness properties of the estimator will be discussed, along with a method for finite-sample bias correction and applications to some real data examples.
