  We consider the problem of performing prediction when observed values are at
their highest levels. We construct an inner product space of nonnegative random
variables from transformed-linear combinations of independent regularly varying
random variables. The matrix of inner products corresponds to the tail pairwise
dependence matrix, which summarizes tail dependence. The projection theorem
yields the optimal transformed-linear predictor, which has the same form as the
best linear unbiased predictor in non-extreme prediction. We also construct
prediction intervals based on the geometry of regular variation. We show that
these intervals have good coverage in a simulation study as well as in two
applications; prediction of high pollution levels, and prediction of large
financial losses.
