  In this article third and fourth order efficiency are studied in the
framework of translation equivariant location estimators. We assume
$X_1,...,X_n$ i.i.d. $f(\cdot -\theta)$. By recognizing that equality in a
special form of the Cauchy-Schwarz inequality leads to a certain dependence of
the cumulants of the maximum likelihood estimator (MLE) for $\theta$, it is
shown that this MLE is fourth order efficient if the underlying distribution is
Gumbel. Contrary to similar results which were previously published this result
is not based on symmetry.
