Modelling the tails of distributions is important in many fields, such as environmental science, hydrology, insurance, engineering and finance, where the risk of unusually large or small events are of interest. This thesis applies extreme value models in neonatal and finance studies and develops novel extreme value modelling for financial applications, to overcome issues associated with the dependence induced by volatility clustering and threshold choice. The instability of preterm infants stimulates the interests in estimating the underlying variability of the physiology measurements typically taken on neonatal intensive care patients. The stochastic volatility model (SVM), fitted using Bayesian inference and a particle filter to capture the on-line latent volatility of oxygen concentration, is used in estimating the variability of medical measurements of preterm infants to highlight instabilities resulting from their under-developed biological systems. Alternative volatility estimators are considered to evaluate the performance of the SVM estimates, the results of which suggest that the stochastic volatility model provides a good estimator of the variability of the oxygen concentration data and therefore may be used to estimate the instantaneous latent volatility for the physiological measurements of preterm infants. The classical extreme value distribution, generalized pareto distribution (GPD), with the peaks-over-threshold (POT) method to ameliorate the impact of dependence in the extremes to infer the extreme quantile of the SVM based variability estimates. Financial returns typically show clusters of observations in the tails, often termed “volatility clustering” which creates challenges when applying extreme value models, since classical extreme value theory assume independence of underlying process. Explicit modelling on GARCH-type dependence behaviour of extremes is developed by implementing GARCH conditional variance structure via the extreme value model parameters. With the combination of GEV and GARCH models, both simulation and empirical results show that the combined model is better suited to explain the extreme quantiles. Another important benefit of the proposed model is that, as a one stage model, it is advantageous in making inferences and accounting for all uncertainties much easier than the traditional two stage approach for capturing this dependence. To tackle the challenge threshold choice in extreme value modelling and the generally asymmetric distribution of financial data, a two tail GPD mixture model is proposed with Bayesian inference to capture both upper and lower tail behaviours simultaneously. The proposed two tail GPD mixture modelling approach can estimate both thresholds, along with other model parameters, and can therefore account for the uncertainty associated with the threshold choice in latter inferences. The two tail GPD mixture model provides a very flexible model for capturing all forms of tail behaviour, potentially allowing for asymmetry in the distribution of two tails, and is demonstrated to be more applicable in financial applications than the one tail GPD mixture models previously proposed in the literature. A new Value-at-Risk (VaR) estimation method is then constructed by adopting the proposed mixture model and two-stage method: where volatility estimation using a latent volatility model (or realized volatility) followed by the two tail GPD mixture model applied to independent innovations to overcome the key issues of dependence, and to account for the uncertainty associated with threshold choice. The proposed method is applied in forecasting VaR for empirical return data during the current financial crisis period.

Cette thèse se place dans le cadre de la Statistique des valeurs extrêmes. Elle y apporte trois contributions principales. L'estimation des quantiles extrêmes se fait dans la littérature en deux étapes. La première étape consiste à utiliser une approximation des quantiles basée sur la théorie des valeurs extrêmes. La deuxième étape consiste à estimer les paramètres inconnus de l'approximation en question, et ce en utilisant les valeurs les plus grandes du jeu de données. Cette décomposition mène à deux erreurs de nature différente, la première étant une erreur systémique de modèle, dite d'approximation ou encore d'extrapolation, la seconde consituant une erreur d'estimation aléatoire. La première contribution de cette thèse est l'étude théorique de cette erreur d'extrapolation mal connue.Cette étude est menée pour deux types d'estimateur différents, tous deux cas particuliers de l'approximation dite de la "loi de Pareto généralisée" : l'estimateur Exponential Tail dédié au domaine d'attraction de Gumbel et l'estimateur de Weissman dédié à celui de Fréchet.Nous montrons alors que l'erreur en question peut s'interpréter comme un reste d'ordre un d'un développement de Taylor. Des conditions nécessaires et suffisantes sont alors établies de telle sorte que l'erreur tende vers zéro quand la taille de l'échantillon augmente. De manière originale, ces conditions mènent à une division du domaine d'attraction de Gumbel en trois parties distinctes. En comparaison, l'erreur d'extrapolation associée à l'estimateur de Weissman présente un comportement unifié sur tout le domaine d'attraction de Fréchet. Des équivalents de l'erreur sont fournis et leur comportement est illustré numériquement. La deuxième contribution est la proposition d'un nouvel estimateur des quantiles extrêmes. Le problème est abordé dans le cadre du modèle ``log Weibull-tail'' généralisé, où le logarithme de l'inverse du taux de hasard cumulé est supposé à variation régulière étendue. Après une discussion sur les conséquences de cette hypothèse, nous proposons un nouvel estimateur des quantiles extrêmes basé sur ce modèle. La normalité asymptotique dudit estimateur est alors établie et son comportement en pratique est évalué sur données réelles et simulées.La troisième contribution de cette thèse est la proposition d'outils permettant en pratique de quantifier les limites d'extrapolation d'un jeu de données. Dans cette optique, nous commençons par proposer des estimateurs des erreurs d'extrapolation associées aux approximations Exponential Tail et Weissman. Après avoir évalué les performances de ces estimateurs sur données simulées, nous estimons les limites d'extrapolation associées à deux jeux de données réelles constitués de mesures journalières de variables environnementales. Dépendant de l'aléa climatique considéré, nous montrons que ces limites sont plus ou moins contraignantes. / This thesis takes place in the extreme value statistics framework. It provides three main contributions to this area. The extreme quantile estimation is a two step approach. First, it consists in proposing an extreme value based quantile approximation. Then, estimators of the unknown quantities are plugged in the previous approximation leading to an extreme quantile estimator.The first contribution of this thesis is the study of this previous approximation error. These investigations are carried out using two different kind of estimators, both based on the well-known Generalized Pareto approximation: the Exponential Tail estimator dedicated to the Gumbel maximum domain of attraction and the Weissman estimator dedicated to the Fréchet one.It is shown that the extrapolation error can be interpreted as the remainder of a first order Taylor expansion. Necessary and sufficient conditions are then provided such that this error tends to zero as the sample size increases. Interestingly, in case of the so-called Exponential Tail estimator, these conditions lead to a subdivision of Gumbel maximum domain of attraction into three subsets. In constrast, the extrapolation error associated with Weissmanestimator has a common behavior over the whole Fréchet maximum domain of attraction. First order equivalents of the extrapolation error are thenderived and their accuracy is illustrated numerically.The second contribution is the proposition of a new extreme quantile estimator.The problem is addressed in the framework of the so-called ``log-Generalized Weibull tail limit'', where the logarithm of the inverse cumulative hazard rate function is supposed to be of extended regular variation. Based on this model, a new estimator of extreme quantiles is proposed. Its asymptotic normality is established and its behavior in practice is illustrated on both real and simulated data.The third contribution of this thesis is the proposition of new mathematical tools allowing the quantification of extrapolation limits associated with a real dataset. To this end, we propose estimators of extrapolation errors associated with the Exponentail Tail and the Weissman approximations. We then study on simulated data how these two estimators perform. We finally use these estimators on real datasets to show that, depending on the climatic phenomena,the extrapolation limits can be more or less stringent.