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  \textbf{\textsc{STANFORD UNIVERSITY}}\\[5pt]
  \textbf{\textsc{DEPARTMENT OF STATISTICS}}\\[5pt]
  \Large{\textbf\textsc{{DEPARTMENTAL SEMINAR}}}
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  4:15 p.m., Tuesday, August 23, 2005\\
  Sequoia Hall Room 200\\
  (Cookies at 3:45 in 1st Floor Lounge)
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  \textsl{Kaspar Rufibach, joint work with Lutz D\"umbgen} \\
  University of Bern, Switzerland \\
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  \textbf{Log-concave density estimation}
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We study nonparametric maximum likelihood estimation of a
univariate log-concave probability density. Many parametric models
feature log-concavity (at least for certain parameter ranges),
e.g. normal, gamma, extreme value, laplace or logistic. Note
further that log-concavity entails uni-modality.

The maximum likelihood estimator can be constructed to be the
solution of a linearly constrained optimization problem. We
provide algorithms using interior point and other methods to solve
this optimization task.

Some general properties and two entirely different
characterizations of the density estimator $\hat f$ are derived.
It is shown that its rate of convergence with respect to supremum
norm on a compact interval $T$ is at least $(\log(n)/n)^{1/3}$ and
typically $(\log(n)/n)^{2/5}$.

This entails that the distribution function estimator $\hat F(x)
:= \int_{-\infty}^x \hat f(t) \; d t$ is essentially equivalent to
the empirical cumulative distribution function.

Furthermore, a general property of log-concave densities enables
one to define a simple plug-in monotone hazard rate estimator
$\hat \lambda$.

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