On the Optimality of Estimating the Tail Index and a Naive Estimator

Using a straightforward estimator for estimating the tail index of a distribution we illustrate the inherent difficulties of this problem. We prove strong and weak consistencies and central limit theorems for our naive estimator, and discuss its various rates of convergence under different conditions. We argue that, while optimal rates of convergence do exist under various conditions for a number of estimators of the tail index, the notion of an optimal sequence for this problem is bound to run into unsurmountable difficulties.