A semiparametric probability distribution estimator of sample maximums

Abstract: Several approaches of nonparametric inference for extreme values have been studied. This study surveys the semiparametric probability distribution estimation of sample maximums. Moriyama (2021) clarified that the parametric fitting to the generalized extreme value distribution becomes large as the tail becomes light, which means the convergence becomes slow. Moriyama (2021) proposed a nonparametric distribution estimator without the fitting of the distribution and obtained asymptotic properties. The nonparametric estimator was proved to outperform the parametrically fitting estimator for light-tailed data. Moreover, it was demonstrated that the parametric fitting estimator numerically outperformed the nonparametric one in other cases. Motivated by the study, we construct two types of semiparametric distribution estimators of sample maximums. The proposed distribution estimators are constructed by mixing the two distribution estimators presented in Moriyama (2021). The cross-validation method and the maximum-likelihood method are presented as a way of estimating the optimal mixing ratio. Simulation experiments clarify the numerical properties of the two types of semiparametric distribution estimators.