  The extreme value index is a fundamental parameter in univariate Extreme
Value Theory (EVT). It captures the tail behavior of a distribution and is
central in the extrapolation beyond observed data. Among other semi-parametric
methods (such as the popular Hill's estimator), the Block Maxima (BM) and
Peaks-Over-Threshold (POT) methods are widely used for assessing the extreme
value index and related normalizing constants. We provide asymptotic theory for
the maximum likelihood estimators (MLE) based on the BM method. Our main result
is the asymptotic normality of the MLE with a non-trivial bias depending on the
extreme value index and on the so-called second order parameter. Our approach
combines asymptotic expansions of the likelihood process and of the empirical
quantile process of block maxima. The results permit to complete the comparison
of most common semi-parametric estimators in EVT (MLE and probability weighted
moment estimators based on the POT or BM methods) through their asymptotic
variances, biases and optimal mean square errors.
