Which Extreme Values are Really Extremes?

The aim of this paper is to give a formal definition and consistent estimates of the extremes of a population. This definition relies on a threshold value that delimits the extremes and on the uniform convergence of the distribution of these extremes to a Pareto type distribution. The tail parameter of this Pareto type distribution is the tail index of the data distribution. The estimator of the threshold is anchored in the Kolmogorov-Smirnov distance between consistent estimates of those two distributions. Our estimator is consistent and via the construction of confidence intervals for the tail index (derived from our threshold estimator) we overcome the bias problems of the usual tail index estimators (Hill or Pickands). The paper also explores the validity of our definition for standard sample sizes. For this purpose, a hypothesis test is designed in order to reject extremes estimates that are not really extremes. Applications for different stock returns are presented