  Hydrological post-processing using quantile regression algorithms constitutes
a prime means of estimating the uncertainty of hydrological predictions.
Nonetheless, conventional large-sample theory for quantile regression does not
apply sufficiently far in the tails of the probability distribution of the
dependent variable. To overcome this limitation that could be crucial when the
interest lies on flood events, hydrological post-processing through extremal
quantile regression is introduced here for estimating the extreme quantiles of
hydrological model's responses. In summary, the new hydrological
post-processing method exploits properties of the Hill's estimator from the
extreme value theory to extrapolate quantile regression's predictions to high
quantiles. As a proof of concept, the new method is here tested in
post-processing daily streamflow simulations provided by three process-based
hydrological models for 180 basins in the contiguous United States (CONUS) and
is further compared to conventional quantile regression. With this large-scale
comparison, it is demonstrated that hydrological post-processing using
conventional quantile regression severely underestimates high quantiles (at the
quantile level 0.9999) compared to hydrological post-processing using extremal
quantile regression, although both methods are equivalent at lower quantiles
(at the quantile level 0.9700). Moreover, it is shown that, in the same
context, extremal quantile regression estimates the high predictive quantiles
with efficiency that is, on average, equivalent in the large-sample study for
the three process-based hydrological models.
