| DATE: | Friday, October 18, 2002 |
| TIME: | TBA |
| PLACE: | LN 2205 |
| SPEAKER: | Erich Haeusler (University of Giessen, Germany) |
| TITLE: | Edgeworth expensions on the Hill estimator in extreme value theory |
Let $F$ be a distribution function with positive extreme value index $\gamma$. The Hill estimator $H_n(k_n)$ based on the upper $k_n+1$ order statistics of a sample of size $n$ from $F$ is one of the most popular estimators for $\gamma$. It is known to be consistent for every intermediate sequence $k_n$, i.e. for every sequence $k_n$ of positive integers satisfying $k_n\to\infty$ and $k_n/n\to 0$ as $n\to\infty$. Asymptotic normality has been studied extensively with the following result for the normalized estimator $k_n^{1/2}(H_n(k_n)-\gamma)$: Under an appropriate second order condition on the upper tail of $F$ for any fixed intermediate sequence $k_n$ the limiting distibution is either normal with mean zero, normal with non-zero mean, or there is no limit distribution at all. Information about the accuracy of the normal approximation, however, seems to be available only through the work of Cheng and Pan (1998) who derived a one-term Edgeworth expansion for the distribution function of $k_n^{1/2}(H_n(k_n)-\gamma)$. Due to their condition on $k_n$ the result covers only part of the range of $k_n$-sequences for which the limit distribution has mean zero. We will show how to extend their approach to cover the full range of $k_n$-sequences and also to allow for higher-order expansions. For this, the size of the bias of $H_n(k_n)$ is the crucial quantity. Moreover, we will also consider the range of $k_n$-sequences for which the normal limit distribution has non-zero mean. It turns out that under third order conditions on the upper tail of $F$ Edgeworth expansions are possible in this range, too.