Model-based inference of conditional extreme value distributions with hydrological applications

Abstract: 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.