  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 $\gamma$.
The convergence rate of the approximation gets slower as $\gamma$ 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
$\gamma> -1$. A numerical comparative study on the bandwidth selection shows
the performances of a plug-in approach and cross-validation approach depend on
$\gamma$ 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 $\gamma$ close to zero as $m$ gets large.
