  Extreme value theory has constructed asymptotic properties of the sample
maximum. This study concerns probability distribution estimation of the sample
maximum. The traditional approach is parametric fitting to the limiting
distribution -- the generalized extreme value distribution; however, the model
in non-limiting cases is misspecified to a certain extent. We propose a plug-in
type of nonparametric estimator that does not need model specification.
Asymptotic properties of the distribution estimator are derived. The simulation
study numerically investigates the relative performance in finite-sample cases.
  This study assumes that the underlying distribution of the original sample
belongs to one of the Hall class, the Weibull class or the bounded class, whose
types of the limiting distributions are all different: the Frechet, Gumbel or
Weibull. It is proven that the convergence rate of the parametric fitting
estimator depends on both the extreme value index and the second-order
parameter and gets slower as the extreme value index tends to zero. On the
other hand, the rate of the nonparametric estimator is proven to be independent
of the extreme value index under certain conditions. The numerical performances
of the parametric fitting estimator and the nonparametric estimator are
compared, which shows that the nonparametric estimator performs better,
especially for the extreme value index close to zero. Finally, we report two
real case studies: the Potomac River peak stream flow (cfs) data and the Danish
Fire Insurance data.
