Parametric and nonparametric probability distribution estimators of sample maximum

TL;DR

This paper compares parametric and nonparametric methods for estimating the distribution of the sample maximum, showing the nonparametric approach often performs better, especially when the extreme value index is near zero.

Contribution

It introduces a nonparametric plug-in estimator for the sample maximum distribution and analyzes its asymptotic properties, comparing it with traditional parametric fitting.

Findings

Nonparametric estimator's convergence rate is independent of the extreme value index.

Nonparametric estimator outperforms parametric fitting near zero extreme value index.

Simulation and real data studies validate the theoretical results.

Abstract

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.