A semiparametric probability distribution estimator of sample maximums

This study proposes a computationally efficient semiparametric distribution estimator, which is a slight modification of the naive mixture proposed by Schuster and Yakowitz (1985) and Olkin and Spiegelman (1987). The proposed method is applied to probability distribution estimation of a sample maximum. Two approaches for the sample maximum distribution estimation, one based on extreme value theory and the other on nonparametric smoothing, exist; however, theoretical and numerical properties of the two approaches are known to heavily depend on the case and greatly differ. This study demonstrates that the semiparametric mixture distribution estimators have good properties of both approaches. The cross-validation method is proposed for the mixing ratio selection for the proposed mixture distribution estimator. The result of simulation experiments and three case studies are reported.