  Multivariate extreme value models are used to estimate joint risk in a number
of applications, with a particular focus on environmental fields ranging from
climatology and hydrology to oceanography and seismic hazards. The
semi-parametric conditional extreme value model of Heffernan and Tawn (2004)
involving a multivariate regression provides the most suitable of current
statistical models in terms of its flexibility to handle a range of extremal
dependence classes. However, the standard inference for the joint distribution
of the residuals of this model suffers from the curse of dimensionality since
in a $d$-dimensional application it involves a $d-1$-dimensional non-parametric
density estimator, which requires, for accuracy, a number points and
commensurate effort that is exponential in $d$. Furthermore, it does not allow
for any partially missing observations to be included and a previous proposal
to address this is extremely computationally intensive, making its use
prohibitive if the proportion of missing data is non-trivial. We propose to
replace the $d-1$-dimensional non-parametric density estimator with a
model-based copula with univariate marginal densities estimated using kernel
methods. This approach provides statistically and computationally efficient
estimates whatever the dimension, $d$ or the degree of missing data. Evidence
is presented to show that the benefits of this approach substantially outweigh
potential mis-specification errors. The methods are illustrated through the
analysis of UK river flow data at a network of 46 sites and assessing the
rarity of the 2015 floods in north west England.
