A composite likelihood consists of a combination of valid likelihood objects, and in particular it is of typical interest to adopt lower dimensional marginal likelihoods. Composite marginal likelihood appears to be an attractive alternative for modeling complex data, and has received increasing attention in handling high dimensional data sets when the joint distribution is computationally difficult to evaluate, or intractable due to complex structure of dependence. We present some aspects of methodological development in composite likelihood inference. The resulting estimator enjoys desirable asymptotic properties such as consistency and asymptotic normality. Composite likelihood based test statistics and their asymptotic distributions are summarized. Higher order asymptotic properties of the signed composite likelihood root statistic are explored. Moreover, we aim to compare accuracy and efficiency of composite likelihood estimation relative to estimation based on ordinary likelihood. Analytical and simulation results are presented for different models, which include multivariate normal distributions, times series model, and correlated binary data.

A composite likelihood consists of a combination of valid likelihood objects, and in particular it is of typical interest to adopt lower dimensional marginal likelihoods. Composite marginal likelihood appears to be an attractive alternative for modeling complex data, and has received increasing attention in handling high dimensional data sets when the joint distribution is computationally difficult to evaluate, or intractable due to complex structure of dependence. We present some aspects of methodological development in composite likelihood inference. The resulting estimator enjoys desirable asymptotic properties such as consistency and asymptotic normality. Composite likelihood based test statistics and their asymptotic distributions are summarized. Higher order asymptotic properties of the signed composite likelihood root statistic are explored. Moreover, we aim to compare accuracy and efficiency of composite likelihood estimation relative to estimation based on ordinary likelihood. Analytical and simulation results are presented for different models, which include multivariate normal distributions, times series model, and correlated binary data.

A maioria dos modelos matemáticos desenvolvidos para eventos raros são baseados em modelos probabilísticos para extremos. Embora as ferramentas para modelagem estatística de extremos univariados e multivariados estejam bem desenvolvidas, a extensão dessas ferramentas para modelar extremos espaciais integra uma área de pesquisa em desenvolvimento muito ativa atualmente. A modelagem de máximos sob o domínio espacial, aplicados a dados meteorológicos é importante para a gestão adequada de riscos e catástrofes ambientais nos países que tem a sua economia profundamente dependente do agronegócio. Uma abordagem natural para tal modelagem é a teoria de extremos espaciais e o processo max-stable, caracterizando-se pela extensão de dimensões infinitas da teoria de valores extremos multivariados, podendo-se então incorporar as funções de correlação existentes na geoestatística e consequentemente, verificar a dependência extrema por meio do coeficiente extremo e o madograma. Neste trabalho descreve-se a aplicação de tais processos na modelagem da dependência de máximos espaciais de precipitação máxima mensal do estado do Paraná, com base em séries históricas observadas em estações meteorológicas. Os modelos propostos consideram o espaço euclidiano e uma transformação denominada espaço climático, que permite explicar a presença de efeitos direcionais, resultantes de padrões meteorológicos sinóticos. Essa metodologia baseia-se no teorema proposto por De Haan (1984) e nos modelos de Smith (1990) e de Schlather (2002), verifica-se também o comportamento isotrópico e anisotrópico desses modelos via simulação Monte Carlo. Estimativas são realizadas através da máxima verossimilhança pareada e os modelos são comparados usando-se o Critério de Informação Takeuchi. O algoritmo utilizado no ajuste é bastante rápido e robusto, permitindo-se uma boa eficiência computacional e estatística na modelagem da precipitação máxima mensal, possibilitando-se a modelagem dos efeitos direcionais resultantes de fenômenos ambientais. / The most mathematical models developed for rare events are based on probabilistic models for extremes. Although the tools for statistical modeling of univariate and multivariate extremes are well-developed, the extension of these tools to model spatial extremes data is currently a very active area of research. Modeling of maximum values under the spatial domain, applied to meteorological data is important for the proper management of risks and environmental disasters in the countries where the agricultural sector has great influence on the economy. A natural approach for such modeling is the theory of extreme spatial and max-stable process, characterized by infinite dimensional extension of multivariate extreme value theory, and we can then incorporate the current correlation functions in geostatistics and thus, check the extreme dependence through the extreme coefficient and the madogram. This thesis describes the application of such procedures in the modeling of spatial maximum dependency of monthly maximum rainfall of Paraná State, historical series based on observed meteorological stations. The proposed models consider the Euclidean space and a transformation called climatic space, which makes it possible to explain the presence of directional effects resulting from synoptic weather patterns. This methodology is based on the theorem proposed by De Haan (1984) and Smith (1990) models and Schlather (2002), checking the isotropic and anisotropic behavior these models through Monte Carlo simulation. Estimates are performed using maximum pairwise likelihood and the models are compared using the Takeuchi information criterion. The algorithm used in the fit is very fast and robust, allowing a good statistical and computational efficiency in monthly maximum rainfall modeling, allowing the modeling of directional effects resulting from environmental phenomena.

Les générateurs stochastiques de temps sont des modèles numériques capables de générer des séquences de données climatiques de longueur souhaitée avec des propriétés statistiques similaires aux données observées. Ces modèles sont de plus en plus utilisés en sciences du climat, hydrologie, agronomie. Cependant, peu de générateurs permettent de simuler plusieurs variables, dont les précipitations, en différents sites d’une région. Dans cette thèse, nous proposons un modèle original de générateur stochastique basé sur un champ gaussien multivarié spatio-temporel. Un premier travail méthodologique a été nécessaire pour développer un modèle de covariance croisée entièrement non séparable adapté à la nature spatio-temporelle multivariée des données étudiées. Cette covariance croisée est une généralisation au cas multivarié du modèle non séparable spatio-temporel de Gneiting dans le cas de la famille de Matérn. La démonstration de la validité du modèle et l’estimation de ses paramètres par maximum de vraisemblance par paires pondérées sont présentées. Une application sur des données climatiques démontre l’intérêt de ce nouveau modèle vis-à-vis des modèles existants. Le champ gaussien multivarié permet la modélisation des résidus des variables climatiques (hors précipitation). Les résidus sont obtenus après normalisation des variables par des moyennes et écarts-types saisonniers, eux-mêmes modélisés par des fonctions sinusoïdales. L’intégration des précipitations dans le générateur stochastique nécessite la transformation d’une composante du champ gaussien par une fonction d’anamorphose. Cette fonction d’anamorphose permet de gérer à la fois l’occurrence et l’intensité des précipitations. La composante correspondante du champ gaussien correspond ainsi à un potentiel de pluie, corrélé aux autres variables par la fonction de covariance croisée développée dans cette thèse. Notre générateur stochastique de temps a été testé sur un ensemble de 18 stations réparties en zone à climat méditerranéen (ou proche) en France. La simulation conditionnelle et non conditionnelle de variables climatiques journalières (températures minimales et maximales, vitesse moyenne du vent, rayonnement solaire et précipitation) pour ces 18 stations soulignent les bons résultats de notre modèle pour un certain nombre de statistiques / Stochastic weather generators are numerical models able to simulate sequences of weather data with similar statistical properties than observed data. However, few of them are able to simulate several variables (with precipitation) at different sites from one region. In this thesis, we propose an original model of stochastic generator based on a spatio-temporal multivariate Gaussian random field. A first methodological work was needed to develop a completely non separable cross-covariance function suitable for the spatio-temporal multivariate nature of studied data. This cross-covariance function is a generalization to the multivariate case of spatio-temporal non-separable Gneiting covariance in the case of the family of Matérn. The proof of the validity of the model and the estimation of its parameters by weighted pairwise maximum likelihood are presented. An application on weather data shows the interest of this new model compared with existing models. The multivariate Gaussian random field allows the modeling of weather variables residuals (excluding precipitation). Residuals are obtained after normalization of variables by seasonal means and standard deviations, themselves modeled by sinusoidal functions. The integration of precipitation in the stochastic generator requires the transformation of a component of the Gaussian random field by an anamorphosis function. This anamorphosis function can manage both the occurrence and intensity of precipitation. The corresponding component of the Gaussian random field corresponds to a rain potential, correlated with other variables by the cross-covariance function developed in this thesis. Our stochastic weather generator was tested on a set of 18 stations distributed over the Mediterranean area (or close) in France. The conditional and non-conditional simulation of daily weather variables (maximum and minimum temperature, average wind speed, solar radiation and precipitation) for these 18 stations show good result for a number of statistics.

A maioria dos modelos matemáticos desenvolvidos para eventos raros são baseados em modelos probabilísticos para extremos. Embora as ferramentas para modelagem estatística de extremos univariados e multivariados estejam bem desenvolvidas, a extensão dessas ferramentas para modelar extremos espaciais integra uma área de pesquisa em desenvolvimento muito ativa atualmente. A modelagem de máximos sob o domínio espacial, aplicados a dados meteorológicos é importante para a gestão adequada de riscos e catástrofes ambientais nos países que tem a sua economia profundamente dependente do agronegócio. Uma abordagem natural para tal modelagem é a teoria de extremos espaciais e o processo max-stable, caracterizando-se pela extensão de dimensões infinitas da teoria de valores extremos multivariados, podendo-se então incorporar as funções de correlação existentes na geoestatística e consequentemente, verificar a dependência extrema por meio do coeficiente extremo e o madograma. Neste trabalho descreve-se a aplicação de tais processos na modelagem da dependência de máximos espaciais de precipitação máxima mensal do estado do Paraná, com base em séries históricas observadas em estações meteorológicas. Os modelos propostos consideram o espaço euclidiano e uma transformação denominada espaço climático, que permite explicar a presença de efeitos direcionais, resultantes de padrões meteorológicos sinóticos. Essa metodologia baseia-se no teorema proposto por De Haan (1984) e nos modelos de Smith (1990) e de Schlather (2002), verifica-se também o comportamento isotrópico e anisotrópico desses modelos via simulação Monte Carlo. Estimativas são realizadas através da máxima verossimilhança pareada e os modelos são comparados usando-se o Critério de Informação Takeuchi. O algoritmo utilizado no ajuste é bastante rápido e robusto, permitindo-se uma boa eficiência computacional e estatística na modelagem da precipitação máxima mensal, possibilitando-se a modelagem dos efeitos direcionais resultantes de fenômenos ambientais. / The most mathematical models developed for rare events are based on probabilistic models for extremes. Although the tools for statistical modeling of univariate and multivariate extremes are well-developed, the extension of these tools to model spatial extremes data is currently a very active area of research. Modeling of maximum values under the spatial domain, applied to meteorological data is important for the proper management of risks and environmental disasters in the countries where the agricultural sector has great influence on the economy. A natural approach for such modeling is the theory of extreme spatial and max-stable process, characterized by infinite dimensional extension of multivariate extreme value theory, and we can then incorporate the current correlation functions in geostatistics and thus, check the extreme dependence through the extreme coefficient and the madogram. This thesis describes the application of such procedures in the modeling of spatial maximum dependency of monthly maximum rainfall of Paraná State, historical series based on observed meteorological stations. The proposed models consider the Euclidean space and a transformation called climatic space, which makes it possible to explain the presence of directional effects resulting from synoptic weather patterns. This methodology is based on the theorem proposed by De Haan (1984) and Smith (1990) models and Schlather (2002), checking the isotropic and anisotropic behavior these models through Monte Carlo simulation. Estimates are performed using maximum pairwise likelihood and the models are compared using the Takeuchi information criterion. The algorithm used in the fit is very fast and robust, allowing a good statistical and computational efficiency in monthly maximum rainfall modeling, allowing the modeling of directional effects resulting from environmental phenomena.

The estimation of latent variable models with ordinal and continuous, or ranking variables is the research focus of this thesis. The existing estimation methods are discussed and a composite likelihood approach is developed. The main advantages of the new method are its low computational complexity which remains unchanged regardless of the model size, and that it yields an asymptotically unbiased, consistent, and normally distributed estimator. The thesis consists of four papers. The first one investigates the two main formulations of the unrestricted Thurstonian model for ranking data along with the corresponding identification constraints. It is found that the extra identifications constraints required in one of them lead to unreliable estimates unless the constraints coincide with the true values of the fixed parameters. In the second paper, a pairwise likelihood (PL) estimation is developed for factor analysis models with ordinal variables. The performance of PL is studied in terms of bias and mean squared error (MSE) and compared with that of the conventional estimation methods via a simulation study and through some real data examples. It is found that the PL estimates and standard errors have very small bias and MSE both decreasing with the sample size, and that the method is competitive to the conventional ones. The results of the first two papers lead to the next one where PL estimation is adjusted to the unrestricted Thurstonian ranking model. As before, the performance of the proposed approach is studied through a simulation study with respect to relative bias and relative MSE and in comparison with the conventional estimation methods. The conclusions are similar to those of the second paper. The last paper extends the PL estimation to the whole structural equation modeling framework where data may include both ordinal and continuous variables as well as covariates. The approach is demonstrated through an example run in R software. The code used has been incorporated in the R package lavaan (version 0.5-11).