Asymptotic properties of parametric and nonparametric probability density estimators of sample maximum
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Asymptotic properties of three estimators of probability density function of sample maximum f_(m):=mfF^m-1 are derived, where m is a function of sample size n. One of the estimators is the parametrically fitted by the approximating generalized extreme value density function. However, the parametric fitting is misspecified in finite m cases. The misspecification comes from mainly the following two: the difference m and the selected block size k, and the poor approximation f_(m) to the generalized extreme value density which depends on the magnitude of m and the extreme index γ. The convergence rate of the approximation gets slower as γ tends to zero. As alternatives two nonparametric density estimators are proposed which are free from the misspecification. The first is a plug-in type of kernel density estimator and the second is a block-maxima-based kernel density estimator. Theoretical study clarifies the asymptotic convergence rate of the plug-in type estimator is faster than the block-maxima-based estimator when γ> -1. A numerical comparative study on the bandwidth selection shows the performances of a plug-in approach and cross-validation approach depend on γ and are totally comparable. Numerical study demonstrates that the plug-in nonparametric estimator with the estimated bandwidth by either approach overtakes the parametrically fitting estimator especially for distributions with γ close to zero as m gets large.
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