Estimating the tail index or fitting a Pareto tail to the data?

by

Ion Grama
University of South Brittany

The talk will be concerned with the problem of estimating the tail index
of a distribution function.
We shall focus on the well known Hill estimator. It is now commonly
recognized that for finite samples sizes the Hill estimator does not
accurately estimate the quantity it was designed to estimate, the index
of regular variation. The so called "Hill horror plot" is a clear
illustration of this. The aim of the present talk is to give a natural
resolution to the "Hill horror plot" paradox and to "rehabilitate" the
Hill estimator, by looking at the problem from the point of view of
selecting an appropriate Pareto type tail. It turns out that, for
finite  sample  sizes,  the Hill estimator is closer to another
quantity, which can be interpreted
as the fitted Pareto index. We identify the Hill estimator  as the
nonparametric maximum likelihood estimator in a particular
semiparametric setting. We establish its efficiency for estimating the
fitted Pareto index.


Address:
Universite de Bretagne-Sud, Laboratoire SABRES
Campus de Tohannic Rue Yves Mainguy
56000 Vannes, FRANCE
ion.grama@univ-ubs.fr