  Estimation of extreme quantile regions, spaces in which future extreme events
can occur with a given low probability, even beyond the range of the observed
data, is an important task in the analysis of extremes. Existing methods to
estimate such regions are available, but do not provide any measures of
estimation uncertainty. We develop univariate and bivariate schemes for
estimating extreme quantile regions under the Bayesian paradigm that
outperforms existing approaches and provides natural measures of quantile
region estimate uncertainty. We examine the method's performance in controlled
simulation studies. We illustrate the applicability of the proposed method by
analysing high bivariate quantiles for pairs of pollutants, conditionally on
different temperature gradations, recorded in Milan, Italy.
