  Motivated by the growing interest in sound forecast evaluation techniques
with an emphasis on distribution tails rather than average behaviour, we
investigate a fundamental question arising in this context: Can statistical
features of distribution tails be elicitable, i.e. be the unique minimizer of
an expected score? We demonstrate that expected scores are not suitable to
distinguish genuine tail properties in a very strong sense. Specifically, we
introduce the class of max-functionals, which contains key characteristics from
extreme value theory, for instance the extreme value index. We show that its
members fail to be elicitable and that their elicitation complexity is in fact
infinite under mild regularity assumptions. Further we prove that, even if the
information of a max-functional is reported via the entire distribution
function, a proper scoring rule cannot separate max-functional values. These
findings highlight the caution needed in forecast evaluation and statistical
inference if relevant information is encoded by such functionals.
