  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 is
in [-1,0]. Replacing the order statistics in tail index estimators by their
corresponding quantiles from the distribution function that is based on the
estimated log-concave density leads to novel smooth quantile and tail index
estimators. These new estimators aim at estimating the tail index especially in
small samples. Acting as a smoother of the empirical distribution function, the
log-concave distribution function estimator reduces estimation variability to a
much greater extent than it introduces bias. As a consequence, Monte Carlo
simulations demonstrate that the smoothed version of the estimators are well
superior to their non-smoothed counterparts, in terms of mean squared error.
