Holger Drees, Universität des Saarlandes, Saarbrucken


On maximal occupation time estimators of the extreme value
index


Many estimators of the extreme value index $\gamma$ of iid observations
are based on $k$ largest order statistics. It is well known that the 
performance of these estimators strongly depends on the value of $k$.
In the last couple of years some procedures for the adaptive choice
of the number of order statistics were introduced. Alternatively, the 
estimator is plotted against $k$ or $\log k$, and the number $k$ is chosen 
by eye in the region where the plot seems most stable. 


In a new approach we propose to determine for each possible value
$\gamma$ the percentage of time the plot spends in certain
neighborhoods of that value and the to define $\hat \gamma$ as the
maximizer of this occupation time. It is shown that $\hat \gamma$ 
automatically converges at the optimal rate towards the true extreme value
index. Moreover, its limit distribution is established under suitable 
second order conditions. 


The basic idea of this approach also applies to many other
semiparametric estimation problems when the performance of the
estimator depends on a smoothing parameter like a bandwith of a kernel 
type estimator in local curve estimation.