An Example of Real–Life Data Where the Hill Estimator is Correct

Abstract

We assume that the normalized excesses over a fixed threshold are distributed according to a Pareto distribution with unknownshape and scale parameters. If the scale parameter is equal to one, then the famous Hill estimator is the maximum likelihoodestimator and, therefore, the asymptotically optimal estimator for the shape parameter. Otherwise, the performance of theHill estimator can be very bad. We present a real-life data set where is evidence that the 1–parameter modeling can be justified.Within the 2–parameter model, we apply the maximum likelihood estimator and modifications of an estimator proposed in Castilloand Hadi (1997).