Cette thèse s'intéresse à l'estimation de certaines mesures de risque d'une variable aléatoire réelle Y en présence d'une covariable X. Pour cela, on va considérer que le vecteur (X,Y) suit une loi elliptique. Dans un premier temps, on va s'intéresser aux quantiles de Y sachant X=x. On va alors tester d'abord un modèle de régression quantile assez répandu dans la littérature, pour lequel on obtient des résultats théoriques que l'on discutera. Face aux limites d'un tel modèle, en particulier pour des niveaux de quantile dits extrêmes, on proposera une nouvelle approche plus adaptée. Des résultats asymptotiques sont donnés, appuyés par une étude numérique puis par un exemple sur des données réelles. Dans un second chapitre, on s'intéressera à une autre mesure de risque appelée expectile. La structure du chapitre est sensiblement la même que celle du précédent, à savoir le test d'un modèle de régression inadapté aux expectiles extrêmes, pour lesquels on propose une approche méthodologique puis statistique. De plus, en mettant en évidence le lien entre les quantiles et expectiles extrêmes, on s'aperçoit que d'autres mesures de risque extrêmes sont étroitement liées aux quantiles extrêmes. On se concentrera sur deux familles appelées Lp-quantiles et mesures d'Haezendonck-Goovaerts, pour lesquelles on propose des estimateurs extrêmes. Une étude numérique est également fournie. Enfin, le dernier chapitre propose quelques pistes pour traiter le cas où la taille de la covariable X est grande. En constatant que nos estimateurs définis précédemment étaient moins performants dans ce cas, on s'inspire alors de quelques méthodes d'estimation en grande dimension pour proposer d'autres estimateurs. Une étude numérique permet d'avoir un aperçu de leurs performances / This PhD thesis focuses on the estimation of some risk measures for a real random variable Y with a covariate vector X. For that purpose, we will consider that the random vector (X,Y) is elliptically distributed. In a first time, we will deal with the quantiles of Y given X=x. We thus firstly investigate a quantile regression model, widespread in the litterature, for which we get theoretical results that we discuss. Indeed, such a model has some limitations, especially when the quantile level is said extreme. Therefore, we propose another more adapted approach. Asymptotic results are given, illustrated by a simulation study and a real data example.In a second chapter, we focus on another risk measure called expectile. The structure of the chapter is essentially the same as that of the previous one. Indeed, we first use a regression model that is not adapted to extreme expectiles, for which a methodological and statistical approach is proposed. Furthermore, highlighting the link between extreme quantiles and expectiles, we realize that other extreme risk measures are closely related to extreme quantiles. We will focus on two families called Lp-quantiles and Haezendonck-Goovaerts risk measures, for which we propose extreme estimators. A simulation study is also provided. Finally, the last chapter is devoted to the case where the size of the covariate vector X is tall. By noticing that our previous estimators perform poorly in this case, we rely on some high dimensional estimation methods to propose other estimators. A simulation study gives a visual overview of their performances

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.

The main objective of this thesis was to explore issues related to student achievement motivation in the Swedish TIMSS 2003 (Trend in International Mathematics and Science Study) context. The thesis comprises of five empirical papers and a summary. The expectancy-value theory of achievement motivation was used as the general theoretical framework in all empirical papers, and all papers are concerned with construct validation in one form or another. Aspects of student achievement motivation were measured on a task-specific level (motivation to do well on the TIMSS test) and on a domain-specific level (self-concept in and valuing of mathematics and science) and regressed on test performance. The first paper reports the development and validation of scores from an instrument measuring aspects related to student test-taking motivation. It was shown that a number of items in the instrument could be interpreted as a measure of test-taking motivation, and that the test-taking motivation construct was distinct from other related constructs. The second paper related the Swedish students’ ratings of mathematics test-taking motivation to mathematics performance in TIMSS 2003. The students in the sample on average reported that they were well motivated to do their best on the TIMSS mathematics test and their ratings of test-taking motivation were positively but rather weakly related to achievement. In the third and the fourth papers, the internal structure and relation to performance of the mathematics and science self-concept and task value scales used in TIMSS internationally was investigated for the Swedish TIMSS 2003 sample. For mathematics, it was shown that the internationally derived scales were suitable also for the Swedish sample. It was further shown that ratings of self-concept were rather strongly related to mathematics achievement while ratings of mathematics value were basically unrelated to mathematics achievement. For the science subjects, the internal structure of the scales was less simple, and ratings of self-concept and valuing of science were not very strongly related to science achievement. The study presented in the fifth paper used interviews and an open-ended questionnaire item to further investigate student test-taking motivation and perceptions of the TIMSS test. The results mainly corroborated the results from study II. In the introductory part of the thesis, the empirical studies are summarized, contextualized, and discussed. The discussion relates obtained results to theoretical assumptions, applied implications, and to issues of validity in the TIMSS context.

A novel microstructure-sensitive extreme value probabilistic framework is introduced to evaluate material performance/variability for damage evolution processes (e.g., fatigue, fracture, creep). This framework employs newly developed extreme value marked correlation functions (EVMCF) to identify the coupled microstructure attributes (e.g., phase/grain size, grain orientation, grain misorientation) that have the greatest statistical relevance to the extreme value response variables (e.g., stress, elastic/plastic strain) that describe the damage evolution processes of interest. This is an improvement on previous approaches that account for distributed extreme value response variables that describe the damage evolution process of interest based only on the extreme value distributions of a single microstructure attribute; previous approaches have given no consideration of how coupled microstructure attributes affect the distributions of extreme value response. This framework also utilizes computational modeling techniques to identify correlations between microstructure attributes that significantly raise or lower the magnitudes of the damage response variables of interest through the simulation of multiple statistical volume elements (SVE). Each SVE for a given response is constructed to be a statistical sample of the entire microstructure ensemble (i.e., bulk material); therefore, the response of interest in each SVE is not expected to be the same. This is in contrast to computational simulation of a single representative volume element (RVE), which often is untenably large for response variables dependent on the extreme value microstructure attributes. This framework has been demonstrated in the context of characterizing microstructure-sensitive high cycle fatigue (HCF) variability due to the processes of fatigue crack formation (nucleation and microstructurally small crack growth) in polycrystalline metallic alloys. Specifically, the framework is exercised to estimate the local driving forces for fatigue crack formation, to validate these with limited existing experiments, and to explore how the extreme value probabilities of certain fatigue indicator parameters (FIPs) affect overall variability in fatigue life in the HCF regime. Various FIPs have been introduced and used previously as a means to quantify the potential for fatigue crack formation based on experimentally observed mechanisms. Distributions of the extreme value FIPs are calculated for multiple SVEs simulated via the FEM with crystal plasticity constitutive relations. By using crystal plasticity relations, the FIPs can be computed based on the cyclic plastic strain on the scale of the individual grains. These simulated SVEs are instantiated such that they are statistically similar to real microstructures in terms of the crystallographic microstructure attributes that are hypothesized to have the most influence on the extreme value HCF response. The polycrystalline alloys considered here include the Ni-base superalloy IN100 and the Ti alloy Ti-6Al-4V. In applying this framework to study the microstructure dependent variability of HCF in these alloys, the extreme value distributions of the FIPs and associated extreme value marked correlations of crystallographic microstructure attributes are characterized. This information can then be used to rank order multiple variants of the microstructure for a specific material system for relative HCF performance or to design new microstructures hypothesized to exhibit improved performance. This framework enables limiting the (presently) large number of experiments required to characterize scatter in HCF and lends quantitative support to designing improved, fatigue-resistant materials and accelerating insertion of modified and new materials into service.

This paper links two applications of Extreme Value Theory (EVT) to analyze Taiwanese financial markets: 1. computation of Value at Risk (VaR) and Expected Shortfall (ES) 2. estimates of cross-market dependence under extreme events. Daily data from the Taiwan Stock Exchange Capitalization Weight Stock Index (TAIEX) and the foreign exchange rate, USD/NTD, are employed to analyze the behavior of each return and the dependence structure between the foreign exchange market and the equity market. In the univariate case, when computing risk measures, EVT provides us a more accurate way to estimate VaR. In bivariate case, when measuring extremal dependence, the results of whole period data show the extremal dependence between two markets is asymptotically independent, and the analyses of subperiods illustrate that the relation is slightly dependent in specific periods. Therefore, there is no significant evidence that extreme events appeared in one market (the equity market or the foreign exchange market) will affect another in Taiwan.

Extreme Value Theory (EVT) is used to analyze the peak sidelobe level distribution for array element positions with arbitrary probability distributions. Computations are discussed in the context of linear antenna arrays using electromagnetic energy. The results also apply to planar arrays of random elements that can be transformed into linear arrays. / Engineering and Applied Sciences

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本文利用Gravelle and Li (2013)提出之系統重要性指標來衡量國內九家上市金控銀行對於系統風險之貢獻程度。此種衡量方法係將特定銀行之系統重要性定義為該銀行發生危機造成系統風險增加的幅度，並以多變量極值理論進行機率的估算。實證結果顯示：一、系統重要性最高者為第一銀行；最低者為中國信託銀行。其中除中國信託銀行之重要性顯著低於其他銀行外，其餘銀行之系統重要性均無顯著差異。二、經營期間較長之銀行其系統重要性較高；具公股色彩之銀行對於系統風險之貢獻程度平均而言高於民營銀行。三、銀行規模與其對系統風險之貢獻大致呈現正向關係，即規模越大之銀行其重要性越高。在此情況下可能會有銀行大到不能倒的問題發生。四、存放比較低之銀行系統重要性亦較低，而資本適足率與系統重要性間並無明顯關係。 / In this thesis, we apply the measure proposed by Gravelle and Li (2013) to examine the systemic importance of certain Taiwanese banks. The systemic importance is defined as the increase in the systemic risk conditioned on the crash of a particular bank, and is estimated by the multivariate extreme value theory. Our empirical evidence shows that the most systemically important bank is First Commercial Bank, and the CTBC Bank is significantly less important than other banks, while the differences among the remaining banks are not significant. Second, banks established earlier have higher systemic importance; and the contribution to systemic risk of public banks, on average, is higher than the contribution of private banks. Third, we also find out that the size of a bank and its risk contribution have positive relationship. That is, the bigger a bank is, the more important it is. Under this circumstances, the too big to fail problem may occur. Last, the bank which has lower loan-to-deposit ratio will be less systemically important than those with higher ones, while the relation between capital adequacy ratio and systemic importance is unclear.

The thesis has for objective to study new statistical methods to correct temperature predictionsthat may be implemented on the ensemble prediction system (EPS) of Meteo France so toimprove its use for the electric system management, at EDF France. The EPS of Meteo Francewe are working on contains 51 members (forecasts by time-step) and gives the temperaturepredictions for 14 days. The thesis contains three parts: in the first one we present the EPSand we implement two statistical methods improving the accuracy or the spread of the EPS andwe introduce criteria for comparing results. In the second part we introduce the extreme valuetheory and the mixture models we use to combine the model we build in the first part withmodels for fitting the distributions tails. In the third part we introduce the quantile regressionas another way of studying the tails of the distribution.