  We discuss the use of likelihood asymptotics for inference on risk measures
in univariate extreme value problems, focusing on estimation of high quantiles
and similar summaries of risk for uncertainty quantification. We study whether
higher-order approximation based on the tangent exponential model can provide
improved inferences, and conclude that inference based on maxima is generally
robust to mild model misspecification and that profile likelihood-based
confidence intervals will often be adequate, whereas inferences based on
threshold exceedances can be badly biased but may be improved by higher-order
methods, at least for moderate sample sizes. We use the methods to shed light
on catastrophic rainfall in Venezuela, flooding in Venice, and the lifetimes of
Italian semi-supercentenarians.
