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 Hammersley- 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 Karamata's 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 users' behavior by a graph.

Ce travail de thèse porte sur les outils statistiques pour l'évaluation des maxima de charges de sloshing dans les cuves de méthaniers. Selon les caractéristiques du navire, son chargement et les conditions de navigation, un ballotement hydrodynamique est observé à l'intérieur des cuves, phénomène communément appelé sloshing. La détermination des charges qui s'appliquent à la structure est basée sur des mesures de pression d'impact au moyen d'essais sur maquette. Les maxima de pression par impact, extraits des mesures, sont étudiés. La durée d'un essai est équivalente à 5 heures au réel et insuffisante pour déterminer des maxima de pression associés à de grandes périodes de retour (40 ans). Un modèle probabiliste est nécessaire pour extrapoler les maxima de pression. Le modèle usuel est une loi de Weibull. Comme ce sont les valeurs extrêmes des échantillons qui nous intéressent, les ajustements sont aussi effectués par les lois des valeurs extrêmes et de Pareto généralisées via les méthodes de maximum par bloc et d'excès au-dessus d'un seuil.L'originalité du travail repose sur l'emploi d'un système alternatif, plus pertinent pour la capture des maxima de pression et d'une quantité de 480 heures de mesures disponible pour les mêmes conditions d'essai. Cela fournit une distribution de référence pour les maxima de pression et nous permet d'évaluer la pertinence des modèles sélectionnés. Nous insistons sur l'importance d'évaluer la qualité des ajustements par des tests statistiques et de quantifier les incertitudes sur les estimations obtenues. La méthodologie fournie a été implémentée dans un logiciel nommé Stat_R qui facilite la manipulation et le traitement des résultats. / This thesis focuses on statistical tools for the assessment of maxima sloshing loads in LNG tanks. According to ship features, tank cargo and sailing conditions, a sloshing phenomenon is observed inside LNG tanks. The determination of sloshing loads supported by the tank structure is derived from impact pressure measurements performed on a test rig. Pressure maxima per impact, extracted from test measurements, are investigated. Test duration is equivalent to 5 hours in full scale. This duration is not sufficient to determine pressure maxima associated with high return periods (40 years). It is necessary to use a probabilistic model in order to extrapolate pressure maxima. Usually, a Weibull model is used. As we focus on extreme values from samples, fittings are also performed with the generalized extreme value distribution and the generalized Pareto distribution using block maximum method and peaks over threshold method.The originality of this work is based on the use of an alternate measurement system which is more relevant than usual measurement system to get pressure maxima and a 480 hours measured data available for same test conditions. This provides a reference distribution for pressure maxima which is used to assess the relevance of the selected probabilistic models. Particular attention is paid to the assessment of fittings quality using statistical tests and to the quantification of uncertainties on estimated values.The provided methodology has been implemented in a software called Stat_R which makes the manipulation and the treatment of results easier.

Department of Statistics / MSc (Statistics) / The ripple e ect of a stock market crash due to extremal dependence is a global issue with key attention and it is at the core of all modelling e orts in risk management. Two methods of extreme value theory (EVT) were used in this study to model equity risk and extremal dependence in the tails of stock market indices from four African emerging markets: South Africa, Nigeria, Kenya and Egypt. The rst is the \bivariate-threshold-excess model" and the second is the \point process approach". With regards to the univariate analysis, the rst nding in the study shows in descending hierarchy that volatility with persistence is highest in the South African market, followed by Egyptian market, then Nigerian market and lastly, the Kenyan equity market. In terms of risk hierarchy, the Egyptian EGX 30 market is the most risk-prone, followed by the South African JSE-ALSI market, then the Nigerian NIGALSH market and the least risky is the Kenyan NSE 20 market. It is therefore concluded that risk is not a brainchild of volatility in these markets. For the bivariate modelling, the extremal dependence ndings indicate that the African continent regional equity markets present a huge investment platform for investors and traders, and o er tremendous opportunity for portfolio diversi cation and investment synergies between markets. These synergistic opportunities are due to the markets being asymptotic (extremal) independent or (very) weak asymptotic dependent and negatively dependent. This outcome is consistent with the ndings of Alagidede (2008) who analysed these same markets using co-integration analysis. The bivariate-threshold-excess and point process models are appropriate for modelling the markets' risks. For modelling the extremal dependence however, given the same marginal threshold quantile, the point process has more access to the extreme observations due to its wider sphere of coverage than the bivariate-threshold-excess model. / NRF