Les modèles stochastiques classiques comportent des copules d'interactions linéaires, exprimant en général des interactions de paire. Il sera envisagé d'étendre ces modèles à des interactions non linéaires de type saturation ou de type triplet, en vue de traiter des applications réalistes, comme les diffusions épidémiques.Le but de cette thèse est d'introduire les fonctions copules en épidémiologie, et surtout d'appliquer ces fonctions sur le système de transmission de la Malaria afin de constater la dépendance entre les différents compartiments du système. Nous étudierons quelques modèles compartimentaux, qui sont une généralisation du modèle de Ross-Macdonald, en supposant que la population n'est pas constante et en prenant en compte des paramètres de transmission comme la fécondité, la mortalité et autres. Aussi, nous introduirons les classes d'âges dans certains de ces modèles compartimentaux, afin de trouver une relation entre les individus de ces classes d'âges à l'aide du modèle de Cox et des fonctions copules. Nous donnerons ensuite, deux exemples sur ces modèles : la Malaria au Mali et la peste en Europe au moyen-âge. Nous introduirons aussi les quantiles conditionnels et les fonctions copules archimédiennes, ce qui nous mènera à trouver une dépendance entre les différents compartiments des hôtes et des vecteurs. / The stochastic classical models include linear interactions copulas, expressing in general pair interactions. It is planned to extend these models to nonlinear interactions of saturation type or triplet type, to treat realistic applications, as the epidemics diffusions.The aim of this thesis is to apply the copulas functions in epidemiology, and especially to apply these functions in the transmission system of malaria to detect the dependence existing between compartments of the epidemic system. We will study some compartmental models, which are a generalization of the Ross-Macdonald model, assuming that the population is not constant and taking into account the transmission parameters such as fertility, mortality, etc. Also, we will introduce the age classes in some of these compartmental models, and study the relationships between individuals of these age classes, using the Cox model and the copulas functions. Then, we will give two examples of these models: the Malaria in Mali and the plague in Europe during the Middle Ages. We will introduce also the conditional quantiles and the Archimedean copulas functions, that will lead us to find dependencies between the different compartments of hosts and vectors.

Philosophiae Doctor - PhD / The major result of this thesis is the development of a framework for the application of pair-mixtures of copulas to model asymmetric dependencies in bivariate data. The main motivation is the inadequacy of mixtures of bivariate Gaussian models which are commonly fitted to data. Mixtures of rotated single parameter Archimedean and Gaussian copulas are fitted to real data sets. The method of maximum likelihood is used for parameter estimation. Goodness-of-fit tests performed on the models giving the highest log-likelihood values show that the models fit the data well. We use mixtures of univariate Gaussian models and mixtures of regression models to investigate the existence of bimodality in the distribution of the widths of autocorrelation functions in a sample of 119 gamma-ray bursts. Contrary to previous findings, our results do not reveal any evidence of bimodality. We extend a study by Genest et al. (2012) of the power and significance levels of tests of copula symmetry, to two copula models which have not been considered previously. Our results confirm that for small sample sizes, these tests fail to maintain their 5% significance level and that the Cramer-von Mises-type statistics are the most powerful.

The objective of quantitative Performance-Based Earthquake Engineering (PBEE) is designing buildings that meet the specified performance objectives when subjected to an earthquake. One challenge to completely relying upon a PBEE approach in design practice is the open-ended nature of characterizing the earthquake ground motion by selecting appropriate ground motions and Intensity Measures (IM) for seismic analysis. This open-ended nature changes the quantified building performance depending upon the ground motions and IMs selected. So, improper ground motion and IM selection can lead to errors in structural performance prediction and thus to poor designs. Hence, the goal of this dissertation is to propose methods and tools that enable an informed selection of earthquake IMs and ground motions, with the broader goal of contributing toward a robust PBEE analysis. In doing so, the change of perspective and the mechanism to incorporate additional information provided by Bayesian methods will be utilized. Evaluation of the ability of IMs towards predicting the response of a building with precision and accuracy for a future, unknown earthquake is a fundamental problem in PBEE analysis. Whereas current methods for IM quality assessment are subjective and have multiple criteria (hence making IM selection challenging), a unified method is proposed that enables rating the numerous IMs. This is done by proposing the first quantitative metric for assessing IM accuracy in predicting the building response to a future earthquake, and then by investigating the relationship between precision and accuracy. This unified metric is further expected to provide a pathway toward improving PBEE analysis by allowing the consideration of multiple IMs. Similar to IM selection, ground motion selection is important for PBEE analysis. Consensus on the "right" input motions for conducting seismic response analyses is often varied and dependent on the analyst. Hence, a general and flexible tool is proposed to aid ground motion selection. General here means the tool encompasses several structural types by considering their sensitivities to different ground motion characteristics. Flexible here means the tool can consider additional information about the earthquake process when available with the analyst. Additionally, in support of this ground motion selection tool, a simplified method for seismic hazard analysis for a vector of IMs is developed. This dissertation addresses four critical issues in IM and ground motion selection for PBEE by proposing: (1) a simplified method for performing vector hazard analysis given multiple IMs; (2) a Bayesian framework to aid ground motion selection which is flexible and general to incorporate preferences of the analyst; (3) a unified metric to aid IM quality assessment for seismic fragility and demand hazard assessment; (4) Bayesian models for capturing heteroscedasticity (non-constant standard deviation) in seismic response analyses which may further influence IM selection. / Doctor of Philosophy / Earthquake ground shaking is a complex phenomenon since there is no unique way to assess its strength. Yet, the strength of ground motion (shaking) becomes an integral part for predicting the future earthquake performance of buildings using the Performance-Based Earthquake Engineering (PBEE) framework. The PBEE framework predicts building performance in terms of expected financial losses, possible downtime, the potential of the building to collapse under a future earthquake. Much prior research has shown that the predictions made by the PBEE framework are heavily dependent upon how the strength of a future earthquake ground motion is characterized. This dependency leads to uncertainty in the predicted building performance and hence its seismic design. The goal of this dissertation therefore is to employ Bayesian reasoning, which takes into account the alternative explanations or perspectives of a research problem, and propose robust quantitative methods that aid IM selection and ground motion selection in PBEE The fact that the local intensity of an earthquake can be characterized in multiple ways using Intensity Measures (IM; e.g., peak ground acceleration) is problematic for PBEE because it leads to different PBEE results for different choices of the IM. While formal procedures for selecting an optimal IM exist, they may be considered as being subjective and have multiple criteria making their use difficult and inconclusive. Bayes rule provides a mechanism called change of perspective using which a problem that is difficult to solve from one perspective could be tackled from a different perspective. This change of perspective mechanism is used to propose a quantitative, unified metric for rating alternative IMs. The immediate application of this metric is aiding the selection of the best IM that would predict the building earthquake performance with least bias. Structural analysis for performance assessment in PBEE is conducted by selecting ground motions which match a target response spectrum (a representation of future ground motions). The definition of a target response spectrum lacks general consensus and is dependent on the analysts’ preferences. To encompass all these preferences and requirements of analysts, a Bayesian target response spectrum which is general and flexible is proposed. While the generality of this Bayesian target response spectrum allow analysts select those ground motions to which their structures are the most sensitive, its flexibility permits the incorporation of additional information (preferences) into the target response spectrum development. This dissertation addresses four critical questions in PBEE: (1) how can we best define ground motion at a site?; (2) if ground motion can only be defined by multiple metrics, how can we easily derive the probability of such shaking at a site?; (3) how do we use these multiple metrics to select a set of ground motion records that best capture the site’s unique seismicity; (4) when those records are used to analyze the response of a structure, how can we be sure that a standard linear regression technique accurately captures the uncertainty in structural response at low and high levels of shaking?

Title: A study of applying copulas in data mining Author: Martin Ščavnický Department: Department of Theoretical Computer Science and Mathe- matical Logic Supervisor: RNDr. Ing. Martin Holeňa CSc., Department of Theoretical Computer Science and Mathematical Logic Abstract: Copulas are functions that describe the relationship between a multivariate distribution function and its marginals. They provide a way to model multivariate distribution functions, and are extensively used in finance and studied in data mining. In practice, there are many different copula families and no standard way for choosing the right one. In our work, we compare suitability of different copula families in data mining. We fit classification data using 8 copula families and compare them using 3 mea- sures of fit. We also use a classification algorithm based on copulas and compare its accuracy for different copula families. The results indicate that elliptical copulas fit our data better, but hierarchical Archimedean copulas give comparable accuracy in the classification. We also propose and test a modified method for modelling data using hierarchical Archimedean copu- las, which fits some datasets with negative dependence between attributes better. Based on this modified method, we propose a visualization of depen- dence in data and observe...

[pt] A representação e interpretação claras da estrutura de dependência presente em vetores aleatórios, em particular em vetores bivariados, podem ser feitas com o uso do conceito de cópulas. Na análise bivariada, os coeficientes de dependência homogênea e heterogênea de cauda têm por objetivo estudar uma medida de dependência quando as variáveis assumem valores extre- mos. Obtemos as expressões dos coeficientes de dependência heterogênea de cauda a partir da função de distribuição acumulada condicional e apresen- tamos a demonstração de que os coeficientes de dependência homogênea de cauda de uma distribuição normal assimétrica são iguais a zero. Com o uso do conceito de cópulas e de dependência de cauda total, estudamos a estru- tura de dependência entre as seguintes variáveis: (i) log-retornos das taxas, interpoladas, para a estrutura a termo pré-fixada de 1 ano e de 2 anos; (ii) log-retorno das taxas para a estrutura a termo pré-fixada de 1 (um) ano e log-retorno do índice do Ibovespa; e (iii) log-retorno das taxas para a estru- tura a termo pré-fixada de 1 (um) ano e log-retorno da expectativa da taxa PTAX, 6 meses a frente. / [en] Using the concepts of copula we can represent and interpret the dependence structure presented in random vectors with clarity, particularly in bivariate vectors. In bivariate analysis, the role of both heterogeneous tail-dependence coefficient and homogenous tail- dependence coefficient are to study a measure of dependence when variables reach extreme values. We find expressions for the heterogeneous tail-dependence coefficients from the conditional cumulative distribution function and prove that the homoge- neous tail-dependence coefficients of a skewed normal distribution are equal to zero. Using the concepts of copula and the total tail dependence, we study the dependence structure between the following variables: (i) log- return of interpolated rates for the 1-year and 2-year fixed term structure; (ii) log-return of interpolated rate for the 1-year and log- return for the Bo- vespa index; e (iii) log-return of interpolated rate for the 1-year fixed term structure and log-return of expected PTAX, 6 months ahead.

Orientador: Armando Zaupa Remacre / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Geociencias / Made available in DSpace on 2018-08-07T09:52:07Z (GMT). No. of bitstreams: 1 Carmo_FredericoAugustoRosado_D.pdf: 2790866 bytes, checksum: 70c1d59f281ee0f7a09af528c73582a9 (MD5) Previous issue date: 2006 / Resumo: O objetivo principal desta tese foi desenvolver a metodologia de cópulas aplicada ao problema de estimativas de reservas condicionadas, corrigindo erros de tonelagem e quantidade de minério de um projeto, via uma abordagem diferente da simulação estocástica condicional. É apresentado um resumo teórico que fundamenta o estudo de cópulas. Inicia-se com a apresentação de definições e conceitos importantes da estatística e da probabilidade. Após uma discussão sobre medidas de correlação, é introduzido o conceito de cópulas, desde sua definição e propriedades básicas até o estudo de alguns tipos de cópulas essenciais para a aplicação nesta tese. É discutida toda a fundamentação teórica desenvolvida para o cálculo de recursos recuperáveis. Os conceitos de curvas de tonelagem e teores são introduzidos, pois são a base da parametrização de reservas minerais. É mostrado como a cópula pode ser utilizada num dos pontos principais da geoestatística mineira, principalmente no que diz respeito ao erro das estimativas. Discorre-se primeiramente sobre o conceito de validação cruzada, apresentando a definição de reserva ilusória, ótima e ideal. É definida a reserva ideal utilizando o conceito de cópulas, onde a krigagem, a simulação seqüencial gaussiana e a cópula são comparadas, mostrando as conseqüências da sobreestimativa e da subestimativa em projetos de cava e seqüenciamento na mineração / Abstract: The aim of this thesis was to develop the applied methodology of copulas in the problem of conditional reserves estimation. The copulas have a different approach from sequential gaussian simulation and in this thesis was used to correct the tonnage and ore quantity of a mining project. It is presented a theoretical summary that is the bases to the study of copulas. It is also' presented a set of definitions and important concepts of the statistics and the probability. After a discussion about correlation measures, is introducing the concept of copulas, begining with the definition and basic properties until the study of some types of essential copulas that was applied in this thesis. Whole the theoretical fundamentation is discussed to developed the calculation of recoverable resources. The concepts of tonnage and grades curves are introduced, therefore they are the base of the parametrization of mineral reserves. It is shown how the copulas can be used in the main points of the mining geostatistics, mainly in what concerns the estimation errors. Firstly the cross validation concept is presented and the illusory, best and ideal reserves are defined. The ideal reserves is defined using the concept of copulas, and the results are compared with the kriging and sequential gaussian simulation. With this comparisons is possible shown the consequences of the upper-estimation and under estimation in an open pit projects and sequential mining layout / Doutorado / Administração e Politica de Recursos Minerais / Doutor em Ciências

Orientador: Celso Kazuyuki Morooka / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica e Instituto de Geociências / Made available in DSpace on 2018-08-22T07:36:07Z (GMT). No. of bitstreams: 1 Filardo_JuarezGuaraci_M.pdf: 9792292 bytes, checksum: fc41bb8bdaac74a7d5124bbc10c3f37b (MD5) Previous issue date: 2012 / Resumo: Este trabalho apresenta a solução para o problema das empresas de petróleo em determinar os quantitativos ótimos de recursos materiais a serem adquiridos para a construção dos poços de petróleo num ambiente de risco, assegurando o cumprimento das metas plurianuais da empresa. Para isto foi necessário gerar vários cenários de demanda pelos materiais através da simulação estocástica do portfólio, pesquisar os mais variados modelos de inventário para entender o dinamismo existente no processo de levantamento de demanda dos materiais até o seu emprego nos poços, possibilitar representar estes fluxos matematicamente por uma função objetivo com termos de receita e de custos destoando dos processos usuais de tomada de decisão que consideram apenas os custos de aquisição; otimizar a função objetivo característica de cada cenário através do método dos algoritmos genéticos para permitir uma flexibilização no uso de modelos mais complexos de representação da demanda, e no final obter uma função multivariada conjunta construída com o auxílio do método das cópulas a qual o tomador de decisão fará o seu julgamento. O processo acima permitiu construir uma metodologia cuja robustez foi testada para um caso prático de definição do quantitativo ótimo de revestimentos de perfuração a ser adquirido para um portfólio de poços constituído por 596 poços marítimos brasileiros em atividades de perfuração, completação, e restauração, e 60 sondas de perfuração com trabalhos simultâneos, intervalo de planejamento de um ano e poços possuindo as mesmas características de projeto do pré-sal. A decisão sobre o quantitativo ideal conforme o risco da empresa foi conseguido facilmente analisando-se as curvas de nível da função multivariada conjunta, que para o caso prático do pré-sal e considerando confiança nos dados de 73%, o quantitativo de revestimento ótimo foi de 140.000 m, caso fosse analisado sem considerar os riscos o quantitativo seria de 145.000 m, indicando que o tomador de decisão tradicional foi conservador e tenderia encarecer o processo. Outra conclusão importante foi que os pontos ótimos situaram-se num intervalo de tempo anterior ao do final do planejamento indicando ter sido vantajoso admitir a falta do revestimento e não construir o poço a partir deste ponto, o que possibilitou também verificar o quão consistente foi o sequenciamento dos poços do portfólio / Abstract: This thesis presents a solution for the problem of oil companies to determine the optimal quantity of material resources to be acquired for the construction of oil wells in a risky environment and assuring the company multiannual goals. In order to achieve that it was necessary to generate various scenarios of materials demand by stochastic simulation of the portfolio, researching the most varied inventory models to understand the existing dynamics in the process of raising materials demand up to their use in the wells, and make possible the representation of these flows mathematically by an objective function in terms of revenue and costs, diverging from the usual decision making processes that consider only the acquisition costs . The characteristic objective function of each scene was optimized by the genetic algorithms method to allow greater flexibility in the use of more complex models to represent the demand by the end of the process is obtained a joint multivariate function built with the aid of the method of copulas and whose decision maker will make their judgment. The above process allowed us to provide a methodology whose robustness has been tested for a practical case of defining the optimum quantity of drilling casings to be acquired for a wells portfolio, consisting of 596 wells in Brazilian offshore drilling, completion, and restore and 60 drilling rigs with simultaneous work, one year for range of planning and wells having the same design features of the sub-salt environment. The decision about the ideal amount considering the company's risk was easily accomplished by analyzing the joint multivariate function contour, to the studied practical case and considering 73% of confidence in the data, the optimum quantity was 140,000 m, if analyzed without considering the risks the quantity would be 145,000 m, indicating that the decision maker was traditionally conservative and would tend to become the process more expensive. Another important conclusion was that the optimal points were located in a time prior to the end of the plan, indicating it was advantageous to allow the lack of the casing and not to build the well from that point, it also allowed to check how consistent the wells portfolio scheduling was / Mestrado / Explotação / Mestre em Ciências e Engenharia de Petróleo

Orientador: Osvair Vidal Trevisan / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica e Instituto de Geociências / Made available in DSpace on 2018-08-27T14:50:44Z (GMT). No. of bitstreams: 1 Marques_JoaoBoscoDias_D.pdf: 8432350 bytes, checksum: 10f225750a03d8f222ff06c7a20cc1e7 (MD5) Previous issue date: 2015 / Resumo: Esta tese, de caráter metodológico, é uma proposta de análise econômica de projetos de petróleo com emprego de cópulas e processos estocásticos autorregressivos envolvendo cinco variáveis fundamentais: o preço da commodity, a taxa mínima de atratividade (TMA), o custo de investimento (CAPEX), o custo operacional (OPEX) e a curva de produção de óleo. A premissa é a existência de uma estratégia de produção já estabelecida, de preferência decorrente de metodologias validadas em simulação numérica de reservatórios. O fluxo de caixa do projeto é baseado numa formulação simplificada de VPL e num modelo analítico de produção condicionado à referida estratégia. Para a aplicação desta metodologia são estimados modelos da família GARCH e ARMA para o preço do óleo e TMA, cópulas Arquimedianas para o CAPEX e o OPEX e cópulas elípticas para as variáveis que compõem a curva analítica de produção. Uma solução computacional, desenvolvida para a validação desta tese, possibilita não só a estimativa dos modelos como a incorporação destes no fluxo de caixa de um projeto de petróleo, tanto em regime de concessão como de partilha de produção. A matriz de incertezas combina os atributos preço e taxa para três cenários econômicos, contra os atributos de eficiência e eficácia para três níveis de desempenho da empresa. A métrica de risco indicada é o valor em risco ponderado. Os resultados evidenciam as vantagens dos modelos estimados para a análise de risco de projetos de petróleo em condições de incertezas. As assimetrias relacionadas aos choques havidos no preço da commodity são claramente evidenciadas. A métrica indicada, além de coerente para um investidor avesso ao risco, pode subsidiar com vantagens a curva de risco na estimativa das reservas P10, P50 e P90. Outra vantagem é o tratamento acoplado de variáveis como o CAPEX e o OPEX como parâmetros de atributos de desempenho operacional. Trata-se de uma metodologia expedita, aplicável, de fácil interpretação e de valor prático, que pode auxiliar os processos de decisão em projetos de alta complexidade como os do pré-sal brasileiro / Abstract: This thesis of methodological nature is a proposed economic analysis of oil projects with the use of copulas and autoregressive stochastic processes involving five fundamental variables: the price of the commodity, the discount rate (TMA), the investment cost (CAPEX), the operating cost (OPEX) and the curve of oil production. The premise is the existence of a production strategy established, preferably derived from validated methods in numerical reservoir simulation. Cash flow of the project is based on a simplified formulation of NPV and an analytical model of production conditioned on the Strategy. For the application of this methodology, models from the ARMA and GARCH family were estimated for the price of oil and TMA, Archimedean copulas for the CAPEX and OPEX, and elliptical copulas for the variables that comprise the analytical production curve. A computational solution, developed to validate this thesis, provides not only the estimation of models, such as the incorporation in the cash flow of an oil project under concession regime and production sharing contract. The uncertainty matrix combines the attributes of price and rate in three economic scenarios versus the attributes of efficiency and effectiveness in three levels of operator performance. The indicated risk metric is the weighted value-at-risk. The results show advantages of the estimated models for the risk analysis of oil projects under conditions of uncertainty. Asymmetries related to shocks that were in the price of commodity are clearly evidenced. The indicated metric, in addition to its consistency for a risk averse investor, can subsidize with advantages the risk curve in the estimation of reserves P10, P50 and P90. Another advantage is the coupled treatment of variables like the CAPEX and OPEX as parameters of management attributes. This is an expedient methodology, applicable, easy to understand and of practical value that can aist decision-making processes in highly complex projects such as Brazilian pre-salt / Doutorado / Reservatórios e Gestão / Doutor em Ciências e Engenharia de Petróleo

This thesis gives an example of assessing the risk of a financial portfolio with international assets, where the assets may be of different classes, by the use of Monte Carlo simulation and Extreme Value Theory. The simulation uses univariate modelling, models of the assets’ returns as stochastic processes, as well as vine copulas to create dependency between the variables. For the asset returns a modified version of a discretized Merton jump diffusion model was used. The risk assessment used Extreme Value Theory to calculate Value at Risk and Expected Shortfall of the simulated portfolio. However, the resulting return distribution, and the risk assessment thereof, was not entirely satisfactory due to unreasonably large values ascertained. / Denna uppsats ger ett exempel på hur riskbedömning av finanisella portföljer med internationella tillgångar av olika tillgångsslag genom Monte Carlo simulering och extremvärdesteori. Simuleringen använder univariat modelling, modeller för tillgångarnas avkastningar som stokastiska processer, såväl som vine-copulas för att skapa ett beroende mellan tillgångarna. Tillgångarnas avkastningar modellerades med en modifierad version av en diskretiserad Merton-jump-diffusion model. Riskbedömningen använde extremvärdesteori för att beräkna Value-at-Risk och Expected-Shortfall. Dock blev den resulterande avkastningsfördelningen och riskbedömningen därav inte helt tillfredsällande på grund av att orimligt stora värden erhölls.

Cette thèse porte sur l'étude de la modélisation et estimation de la dépendance des portefeuilles de risques et l'estimation du risque agrégé. Dans le Chapitre 2, nous proposons une nouvelle méthode pour estimer les quantiles de haut niveau pour une somme de risques. Elle est basée sur l'estimation du rapport entre la VaR de la somme et la VaR du maximum des risques. Nous utilisons des résultats sur les fonctions à variation régulière. Nous comparons l'efficacité de notre méthode avec quelques estimations basées sur la théorie des valeurs extrêmes, sur plusieurs modèles. Notre méthode donne de bons résultats lors de l'approximation de la VaR à des niveaux élevés lorsque les risques sont fortement dépendants et au moins l'un des risques est à queue épaisse. Dans le Chapitre 3, nous proposons une procédure d'estimation pour la distribution d'un risque agrégé basée sur la copule échiquier. Elle permet d'obtenir de bonnes estimations à partir d'un petit échantillon de la loi multivariée et une connaissance complète des lois marginales. Cette situation est réaliste pour de nombreuses applications. Les estimations peuvent être améliorées en incluant dans la copule échiquier des informations supplémentaires (sur la loi d'un sous-vecteur ou sur des probabilités extrêmes). Notre approche est illustrée par des exemples numériques. Finalement, dans le Chapitre 4, nous proposons un estimateur de la mesure spectrale basé sur l'estimation à noyau de la densité de la mesure spectrale d'une distribution à variation régulière bivariée. Une extension de notre méthode permet d'estimer la mesure spectrale discrète. Certaines propriétés de convergence sont obtenues / This thesis comprises three essays on estimation methods for the dependence between risks and its aggregation. In the first essay we propose a new method to estimate high level quantiles of sums of risks. It is based on the estimation of the ratio between the VaR (or TVaR) of the sum and the VaR (or TVaR) of the maximum of the risks. We use results on regularly varying functions. We compare the efficiency of our method with classical ones, on several models. Our method gives good results when approximating the VaR or TVaR in high levels on strongly dependent risks where at least one of the risks is heavy tailed. In the second essay we propose an estimation procedure for the distribution of an aggregated risk based on the checkerboard copula. It allows to get good estimations from a (quite) small sample of the multivariate law and a full knowledge of the marginal laws. This situation is realistic for many applications. Estimations may be improved by including in the checkerboard copula some additional information (on the law of a sub-vector or on extreme probabilities). Our approach is illustrated by numerical examples. In the third essay we propose a kernel based estimator for the spectral measure density of a bivariate distribution with regular variation. An extension of our method allows to estimate discrete spectral measures. Some convergence properties are obtained