  The maximum likelihood method offers a standard way to estimate the three
parameters of a generalized extreme value (GEV) distribution. Combined with the
block maxima method, it is often used in practice to assess the extreme value
index and normalization constants of a distribution satisfying a first order
extreme value condition, assuming implicitely that the block maxima are exactly
GEV distributed. This is unsatisfactory since the GEV distribution is a good
approximation of the block maxima distribution only for blocks of large size.
The purpose of this paper is to provide a theoretical basis for this
methodology. Under a first order extreme value condition only, we prove the
existence and consistency of the maximum likelihood estimators for the extreme
value index and normalization constants within the framework of the block
maxima method.
