Recent years have seen an upsurge of research on the modeling of extreme events, largely driven by impetus from two major areas of application: finance and environmental science. Many problems resulting from these areas are inherently multivariate and therefore should be analyzed with proper multivariate extreme value methods. For instance, in finance an important problem is to estimate very large quantiles of the distribution of the sums of possibly dependent risks. Our research focus is on (conditional) multivariate extremes.
In the paper ‘Local robust estimation of the Pickands dependence function’ (Annals of Statistics, 46, 2806-2843), we consider the robust estimation of the Pickands dependence function in the random covariate framework. Our estimator is based on local estimation with the minimum density power divergence criterion. We provide the main asymptotic properties, in particular the convergence of the stochastic process, correctly normalized, towards a tight centered Gaussian process. The finite sample performance of our estimator is evaluated with a simulation study involving both uncontaminated and contaminated samples.
As a second example, in the publication ‘Local estimation of the conditional stable tail dependence function’ (Scandinavian Journal of Statistics, 45, 590-617), we study the local estimation of the stable tail dependence function when a random covariate is observed together with the variables of main interest. Our estimator is a weighted version of the empirical estimator adapted to the covariate framework. We provide the main asymptotic properties of our estimator, when properly normalized, in particular the convergence of the empirical process towards a tight centered Gaussian process.
The finite sample performance of our above mentioned estimators is illustrated on a data set of air pollution measurements. Being able to analyse the dependence between temperature and ozone concentration is of primary importance in order to identify population health effects of high ozone concentration and extreme temperature. The dataset contains daily measurements on, among others, maximum temperature and ground level ozone concentration, for the time period 1999 to 2013, collected at stations spread over the U.S. by the United States Environmental Protection Agency (EPA). The figure above illustrates the estimate of extreme dependence on January 15, 2007 in California. We discover that the extremal dependence between daily maximum temperature and ground level ozone concentration varies a lot across measurement stations which can be explained by the fact that the climate of California varies widely, from hot desert to subarctic, depending on the location.