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  \textbf{\Large{\textsc{STANFORD UNIVERSITY}}}\\[5pt]
  \textbf{\Large{\textsc{DEPARTMENT OF STATISTICS}}}\\[5pt]
  \Large{\textsc{DEPARTMENTAL SEMINAR}}
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  Example: 4:15 p.m., Tuesday, July 31, 2007\\
  Sequoia Hall Room 200\\
  (Cookies at 3:45 in 1st Floor Lounge)
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  \textsl{Kaspar Rufibach} \\
  Department of Statistics\\
  Stanford University
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  \subsection*{Max-–domain of attraction for log–-concave \\densities and smooth tail index estimation}
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\noindent Both parametric distribution functions appearing in extreme value theory -- the generalized extreme
value distribution and the generalized Pareto distribution -- have log--concave densities if the extreme value
index $\gamma$ lies in $[-1,0]$. It is shown that all distribution functions $F$ having a log--concave density
function belong to the max--domain of attraction of the generalized extreme value distribution.

Given an i.i.d.~sample $X_1, \ldots, X_n$ where $X_i$ has a log--concave density $f$, the distribution function
$\hat F_n$ derived from the log--concave NPMLE $\hat f_n$ is asymptotically equivalent to the empirical
distribution function $\mathbb{F}_n$. Replacing the order statistics in tail index estimators by the quantiles
of $\hat F_n$ leads to ``smoothed'' estimators of $\gamma$. Monte Carlo simulations suggest that for finite $n$
these new estimators are highly accurate and well superior to their non--smoothed counterparts. If time permits,
we discuss some problems in deriving asymptotical results.

Based on joint work with Samuel M\"uller, University of Western Australia.


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