Over the past four decades, we have witnessed an increasing interest in developing and
using multivariate extreme value theory to deal with the growing concern for risk assessment
of rare phenomena, due to their large socio-economic impacts. The extreme value
community have realized the prominence of considering the multivariate nature of extreme
events to take into account the risk component emerging from the dependence between
these events, but have also recognized the additional complexity of this task compared to
the well-understood univariate extreme value theory. Moreover, the non-stationary extent
of most extreme events have focused the attention of many researchers over the past few
years, owing to the auxiliary information that a set of covariates might contain, thereby
allowing data pooling, improved inference, and a better understanding of the behaviour
of extreme events in different environments and settings.
In this thesis, we develop new models for covariate-varying tail dependence structures
based on asymptotically justified arguments, and propose novel techniques for fitting
these models to both block maxima and threshold exceedances data, under the assumptions
of asymptotic dependence and asymptotic independence. Our proposals for the
flexible incorporation of covariate influence on the extremal dependence rely on the (vector)
generalized additive modelling infrastructure, and are established in a parametric
setting where we extend the standard approach of modelling non-stationary univariate
extremes to the multivariate framework through spectral density modelling, as well as
a non-parametric setting where we develop projection techniques enabling the reduction
of the problem of characterizing joint tail dependences to the modelling of univariate
random variables. Inference is performed by penalized maximum likelihood estimation
combined, when applicable, with censored likelihood techniques. The performance of the
resulting estimators is assessed either through simulation studies or based on asymptotic
distributions, when the parametric approach to the extremal dependence modelling is
undertaken.
The developed methodologies are illustrated on environmental datasets where dependence
between large events is linked to a set of covariates describing time as well as
characteristics of the measurement sites.