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Title: Nonparametric distribution estimators of sample maximum in iid settings Authors:  Moriyama Taku - Tottori University (Japan) [presenting]
Abstract: Extreme value theory has constructed asymptotic properties of the sample maximum under some parametric assumptions. The focus is on the 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 finite cases is misspecified to a certain extent. We propose nonparametric estimators that do not need model specification. Asymptotic properties of the distribution estimators are derived. The numerical performances of the parametric estimator and the nonparametric estimators are compared. A simulation experiment demonstrates the obtained asymptotic convergence rates and clarifies the influence of misspecification in the context of probability distribution estimation. It is assumed that the underlying distribution of the original sample belongs to one of the Hall class, the Weibull class and the bounded class, whose types of the limiting distributions are all different: the Frechet, Gumbel and Weibull. It is proven that the convergence rate of the parametric fitting estimator depends on both the tail index and the second-order parameter and gets slow as the tail index tends to zero. The simulation results are generally consistent with the obtained convergence rates. Finally, we report two real case studies: the Potomac River peak stream flow data and the Danish Fire Insurance data.