  We consider forecasting functional time series of extreme values within a
generalised extreme value distribution (GEV). The GEV distribution can be
characterised using the three parameters (location, scale and shape). As a
result, the forecasts of the GEV density can be accomplished by forecasting
these three latent parameters. Depending on the underlying data structure, some
of the three parameters can either be modelled as scalars or functions. We
provide two forecasting algorithms to model and forecast these parameters. To
assess the forecast uncertainty, we apply a sieve bootstrap method to construct
pointwise and simultaneous prediction intervals of the forecasted extreme
values. Illustrated by a daily maximum temperature dataset, we demonstrate the
advantages of modelling these parameters as functions. Further, the
finite-sample performance of our methods is quantified using several
Monte-Carlo simulated data under a range of scenarios.
