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

Asymptotic properties of parametric and nonparametric probability density estimators of sample maximum