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.

翻译：极值理论已构建了样本最大值的渐近性质。本研究关注样本最大值的概率分布估计。传统方法是对极限分布——广义极值分布进行参数拟合；然而，在非极限情形下该模型存在一定程度的设定误差。我们提出一种无需模型设定的插件型非参数估计器。推导了该分布估计器的渐近性质。模拟研究通过数值方法考察了有限样本情形下的相对性能。本研究假设原始样本的基础分布属于Hall类、Weibull类或有界类中的一种，其极限分布类型各不相同：分别为Fréchet分布、Gumbel分布或Weibull分布。研究证明，参数拟合估计器的收敛速率同时依赖于极值指数与二阶参数，且随着极值指数趋近于零而减慢。另一方面，在特定条件下，非参数估计器的速率被证明与极值指数无关。通过比较参数拟合估计器与非参数估计器的数值表现，发现非参数估计器具有更优性能，尤其在极值指数接近零时更为显著。最后，我们报告了两个实际案例研究：波托马克河峰值流量（立方英尺/秒）数据与丹麦火灾保险数据。