Les modèles à sur-représentation de zéros discrets et continus ont une large gamme d'applications et leurs propriétés sont bien connues. Bien qu'il existe des travaux portant sur les modèles discrets à sous-représentation de zéro et modifiés à zéro, la formulation usuelle des modèles continus à sur-représentation -- un mélange entre une densité continue et une masse de Dirac -- empêche de les généraliser afin de couvrir le cas de la sous-représentation de zéros. Une formulation alternative des modèles continus à sur-représentation de zéros, pouvant aisément être généralisée au cas de la sous-représentation, est présentée ici. L'estimation est d'abord abordée sous le paradigme classique, et plusieurs méthodes d'obtention des estimateurs du maximum de vraisemblance sont proposées. Le problème de l'estimation ponctuelle est également considéré du point de vue bayésien. Des tests d'hypothèses classiques et bayésiens visant à déterminer si des données sont à sur- ou sous-représentation de zéros sont présentées. Les méthodes d'estimation et de tests sont aussi évaluées au moyen d'études de simulation et appliquées à des données de précipitation agrégées. Les diverses méthodes s'accordent sur la sous-représentation de zéros des données, démontrant la pertinence du modèle proposé. Nous considérons ensuite la classification d'échantillons de données à sous-représentation de zéros. De telles données étant fortement non normales, il est possible de croire que les méthodes courantes de détermination du nombre de grappes s'avèrent peu performantes. Nous affirmons que la classification bayésienne, basée sur la distribution marginale des observations, tiendrait compte des particularités du modèle, ce qui se traduirait par une meilleure performance. Plusieurs méthodes de classification sont comparées au moyen d'une étude de simulation, et la méthode proposée est appliquée à des données de précipitation agrégées provenant de 28 stations de mesure en Colombie-Britannique. / Zero-inflated models, both discrete and continuous, have a large variety of applications and fairly well-known properties. Some work has been done on zero-deflated and zero-modified discrete models. The usual formulation of continuous zero-inflated models -- a mixture between a continuous density and a Dirac mass at zero -- precludes their extension to cover the zero-deflated case. We introduce an alternative formulation of zero-inflated continuous models, along with a natural extension to the zero-deflated case. Parameter estimation is first studied within the classical frequentist framework. Several methods for obtaining the maximum likelihood estimators are proposed. The problem of point estimation is considered from a Bayesian point of view. Hypothesis testing, aiming at determining whether data are zero-inflated, zero-deflated or not zero-modified, is also considered under both the classical and Bayesian paradigms. The proposed estimation and testing methods are assessed through simulation studies and applied to aggregated rainfall data. The data is shown to be zero-deflated, demonstrating the relevance of the proposed model. We next consider the clustering of samples of zero-deflated data. Such data present strong non-normality. Therefore, the usual methods for determining the number of clusters are expected to perform poorly. We argue that Bayesian clustering based on the marginal distribution of the observations would take into account the particularities of the model and exhibit better performance. Several clustering methods are compared using a simulation study. The proposed method is applied to aggregated rainfall data sampled from 28 measuring stations in British Columbia.

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

Le but de ce mémoire de maîtrise est de décrire les propriétés de la loi double Pareto-lognormale, de montrer comment on peut introduire des variables explicatives dans le modèle et de présenter son large potentiel d'applications dans le domaine de la science actuarielle et de la finance. Tout d'abord, nous donnons la définition de la loi double Pareto-lognormale et présentons certaines de ses propriétés basées sur les travaux de Reed et Jorgensen (2004). Les paramètres peuvent être estimés en utilisant la méthode des moments ou le maximum de vraisemblance. Ensuite, nous ajoutons une variable explicative à notre modèle. La procédure d'estimation des paramètres de ce mo-\\dèle est également discutée. Troisièmement, des applications numériques de notre modèle sont illustrées et quelques tests statistiques utiles sont effectués. / The purpose of this Master's thesis is to describe the double Pareto-lognormal distribution, show how the model can be extended by introducing explanatory variables in the model and present its large potential of applications in actuarial science and finance. First, we give the definition of the double Pareto-lognormal distribution and present some of its properties based on the work of Reed and Jorgensen (2004). The parameters could be estimated by using the method of moments or maximum likelihood. Next, we add an explanatory variable to our model. The procedure of estimation for this model is also discussed. Finally, some numerical applications of our model are illustrated and some useful statistical tests are conducted.

Esta tese visa propor extensões dos modelos de regressão quantílica bayesianos, considerando dados de proporção com inflação de zeros, e também dados censurados no zero. Inicialmente, é sugerida uma análise de observações influentes, a partir da representação por mistura localização-escala da distribuição Laplace assimétrica, em que as distribuições a posteriori das variáveis latentes são comparadas com o intuito de identificar possíveis observações aberrantes. Em seguida, é proposto um modelo de duas partes para analisar dados de proporção com inflação de zeros ou uns, estudando os quantis condicionais e a probabilidade da variável resposta ser igual a zero. Além disso, são propostos modelos de regressão quantílica bayesiana para dados contínuos com um componente discreto no zero, em que parte dessas observações é suposta censurada. Esses modelos podem ser considerados mais completos na análise desse tipo de dados, uma vez que a probabilidade de censura é verificada para cada quantil de interesse. E por último, é considerada uma aplicação desses modelos com correlação espacial, para estudar os dados da eleição presidencial no Brasil em 2014. Nesse caso, os modelos de regressão quantílica são capazes de incorporar essa informação espacial a partir do processo Laplace assimétrico. Para todos os modelos propostos foi desenvolvido um pacote do software R, que está exemplificado no apêndice. / This thesis aims to propose extensions of Bayesian quantile regression models, considering proportion data with zero inflation, and also censored data at zero. Initially, it is suggested an analysis of influential observations, based on the location-scale mixture representation of the asymmetric Laplace distribution, where the posterior distribution of the latent variables are compared with the goal of identifying possible outlying observations. Next, a two-part model is proposed to analyze proportion data with zero or one inflation, studying the conditional quantile and the probability of the response variable being equal to zero. Following, Bayesian quantile regression models are proposed for continuous data with a discrete component at zero, where part of these observations are assumed censored. These models may be considered more complete in the analysis of this type of data, as the censoring probability varies with the quantiles of interest. For last, it is considered an application of these models with spacial correlation, in order to study the data about the last presidential election in Brazil in 2014. In this example, the quantile regression models are able to incorporate spatial dependence with the asymmetric Laplace process. For all the proposed models it was developed a R package, which is exemplified in the appendix.

Les modèles à sur-représentation de zéros discrets et continus ont une large gamme d'applications et leurs propriétés sont bien connues. Bien qu'il existe des travaux portant sur les modèles discrets à sous-représentation de zéro et modifiés à zéro, la formulation usuelle des modèles continus à sur-représentation -- un mélange entre une densité continue et une masse de Dirac -- empêche de les généraliser afin de couvrir le cas de la sous-représentation de zéros. Une formulation alternative des modèles continus à sur-représentation de zéros, pouvant aisément être généralisée au cas de la sous-représentation, est présentée ici. L'estimation est d'abord abordée sous le paradigme classique, et plusieurs méthodes d'obtention des estimateurs du maximum de vraisemblance sont proposées. Le problème de l'estimation ponctuelle est également considéré du point de vue bayésien. Des tests d'hypothèses classiques et bayésiens visant à déterminer si des données sont à sur- ou sous-représentation de zéros sont présentées. Les méthodes d'estimation et de tests sont aussi évaluées au moyen d'études de simulation et appliquées à des données de précipitation agrégées. Les diverses méthodes s'accordent sur la sous-représentation de zéros des données, démontrant la pertinence du modèle proposé. Nous considérons ensuite la classification d'échantillons de données à sous-représentation de zéros. De telles données étant fortement non normales, il est possible de croire que les méthodes courantes de détermination du nombre de grappes s'avèrent peu performantes. Nous affirmons que la classification bayésienne, basée sur la distribution marginale des observations, tiendrait compte des particularités du modèle, ce qui se traduirait par une meilleure performance. Plusieurs méthodes de classification sont comparées au moyen d'une étude de simulation, et la méthode proposée est appliquée à des données de précipitation agrégées provenant de 28 stations de mesure en Colombie-Britannique. / Zero-inflated models, both discrete and continuous, have a large variety of applications and fairly well-known properties. Some work has been done on zero-deflated and zero-modified discrete models. The usual formulation of continuous zero-inflated models -- a mixture between a continuous density and a Dirac mass at zero -- precludes their extension to cover the zero-deflated case. We introduce an alternative formulation of zero-inflated continuous models, along with a natural extension to the zero-deflated case. Parameter estimation is first studied within the classical frequentist framework. Several methods for obtaining the maximum likelihood estimators are proposed. The problem of point estimation is considered from a Bayesian point of view. Hypothesis testing, aiming at determining whether data are zero-inflated, zero-deflated or not zero-modified, is also considered under both the classical and Bayesian paradigms. The proposed estimation and testing methods are assessed through simulation studies and applied to aggregated rainfall data. The data is shown to be zero-deflated, demonstrating the relevance of the proposed model. We next consider the clustering of samples of zero-deflated data. Such data present strong non-normality. Therefore, the usual methods for determining the number of clusters are expected to perform poorly. We argue that Bayesian clustering based on the marginal distribution of the observations would take into account the particularities of the model and exhibit better performance. Several clustering methods are compared using a simulation study. The proposed method is applied to aggregated rainfall data sampled from 28 measuring stations in British Columbia.

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal

Esta tese visa propor extensões dos modelos de regressão quantílica bayesianos, considerando dados de proporção com inflação de zeros, e também dados censurados no zero. Inicialmente, é sugerida uma análise de observações influentes, a partir da representação por mistura localização-escala da distribuição Laplace assimétrica, em que as distribuições a posteriori das variáveis latentes são comparadas com o intuito de identificar possíveis observações aberrantes. Em seguida, é proposto um modelo de duas partes para analisar dados de proporção com inflação de zeros ou uns, estudando os quantis condicionais e a probabilidade da variável resposta ser igual a zero. Além disso, são propostos modelos de regressão quantílica bayesiana para dados contínuos com um componente discreto no zero, em que parte dessas observações é suposta censurada. Esses modelos podem ser considerados mais completos na análise desse tipo de dados, uma vez que a probabilidade de censura é verificada para cada quantil de interesse. E por último, é considerada uma aplicação desses modelos com correlação espacial, para estudar os dados da eleição presidencial no Brasil em 2014. Nesse caso, os modelos de regressão quantílica são capazes de incorporar essa informação espacial a partir do processo Laplace assimétrico. Para todos os modelos propostos foi desenvolvido um pacote do software R, que está exemplificado no apêndice. / This thesis aims to propose extensions of Bayesian quantile regression models, considering proportion data with zero inflation, and also censored data at zero. Initially, it is suggested an analysis of influential observations, based on the location-scale mixture representation of the asymmetric Laplace distribution, where the posterior distribution of the latent variables are compared with the goal of identifying possible outlying observations. Next, a two-part model is proposed to analyze proportion data with zero or one inflation, studying the conditional quantile and the probability of the response variable being equal to zero. Following, Bayesian quantile regression models are proposed for continuous data with a discrete component at zero, where part of these observations are assumed censored. These models may be considered more complete in the analysis of this type of data, as the censoring probability varies with the quantiles of interest. For last, it is considered an application of these models with spacial correlation, in order to study the data about the last presidential election in Brazil in 2014. In this example, the quantile regression models are able to incorporate spatial dependence with the asymmetric Laplace process. For all the proposed models it was developed a R package, which is exemplified in the appendix.

To extract the best possible information from geodetic and geophysical observations, it is necessary to select a model of the observation errors, mostly the family of Gaussian normal distributions. However, there are alternatives, typically chosen in the framework of robust M-estimation. We give a synopsis of well-known and less well-known models for observation errors and propose to select a model based on information criteria. In this contribution we compare the Akaike information criterion (AIC) and the Anderson Darling (AD) test and apply them to the test problem of fitting a straight line. The comparison is facilitated by a Monte Carlo approach. It turns out that the model selection by AIC has some advantages over the AD test.

To extract the best possible information from geodetic and geophysical observations, it is necessary to select a model of the observation errors, mostly the family of Gaussian normal distributions. However, there are alternatives, typically chosen in the framework of robust M-estimation. We give a synopsis of well-known and less well-known models for observation errors and propose to select a model based on information criteria. In this contribution we compare the Akaike information criterion (AIC) and the Anderson Darling (AD) test and apply them to the test problem of fitting a straight line. The comparison is facilitated by a Monte Carlo approach. It turns out that the model selection by AIC has some advantages over the AD test.

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