  Phenomena such as air pollution levels are of greatest interest when
observations are large, but standard prediction methods are not specifically
designed for large observations. We propose a method, rooted in extreme value
theory, which approximates the conditional distribution of an unobserved
component of a random vector given large observed values. Specifically, for
$\mathbf{Z}=(Z_1,...,Z_d)^T$ and $\mathbf{Z}_{-d}=(Z_1,...,Z_{d-1})^T$, the
method approximates the conditional distribution of
$[Z_d|\mathbf{Z}_{-d}=\mathbf{z}_{-d}]$ when $|\mathbf{z}_{-d}|>r_*$. The
approach is based on the assumption that $\mathbf{Z}$ is a multivariate
regularly varying random vector of dimension $d$. The conditional distribution
approximation relies on knowledge of the angular measure of $\mathbf{Z}$, which
provides explicit structure for dependence in the distribution's tail. As the
method produces a predictive distribution rather than just a point predictor,
one can answer any question posed about the quantity being predicted, and, in
particular, one can assess how well the extreme behavior is represented. Using
a fitted model for the angular measure, we apply our method to nitrogen dioxide
measurements in metropolitan Washington DC. We obtain a predictive distribution
for the air pollutant at a location given the air pollutant's measurements at
four nearby locations and given that the norm of the vector of the observed
measurements is large.
