Modelling of extremes

This work focuses on statistical methods to understand how frequently rare events occur and what the magnitude of extreme values such as large losses is. It lies in a field called extreme value analysis whose scope is to provide support for scientific decision making when extreme observations are of particular importance such as in environmental applications, insurance and finance. In the univariate case, I propose new techniques to model tails of discrete distributions and illustrate them in an application on word frequency and multiple birth data. Suitably rescaled, the limiting tails of some discrete distributions are shown to converge to a discrete generalized Pareto distribution and generalized Zipf distribution respectively. In the multivariate high-dimensional case, I suggest modeling tail dependence between random variables by a graph such that its nodes correspond to the variables and shocks propagate through the edges. Relying on the ideas of graphical models, I prove that if the variables satisfy a new notion called asymptotic conditional independence, then the density of the joint distribution can be simplified and expressed in terms of lower dimensional functions. This generalizes the Hammersleyâ Clifford theorem and enables us to infer tail distributions from observations in reduced dimension. As an illustration, extreme river flows are modeled by a tree graphical model whose structure appears to recover almost exactly the actual river network. A fundamental concept when studying limiting tail distributions is regular variation. I propose a new notion in the multivariate case called one-component regular variation, of which Karamataâs and the representation theorem, two important results in the univariate case, are generalizations. Eventually, I turn my attention to website visit data and fit a censored copula Gaussian graphical model allowing the visualization of usersâ behavior by a graph.