  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.
