Conjuntos de dados com respostas multivariadas aparecem frequentemente em pesquisas em que os dados são provenientes de questionários. Exemplos mais comuns são pesquisas de opinião, mais especificamente, pesquisas de marketing em que a preferência do consumidor em potencial é avaliado: pelo produto, marca, preço, praça, promoção e etc. Um tipo pesquisa de opinião que ganha grande destaque no Brasil de dois em dois anos são as pesquisas eleitorais de intenção de votos. Nós introduzimos nesta tese uma classe de modelos de regressão multivariados com funções de ligação assimétricas para o ajuste de conjuntos de dados com respostas multivariadas binárias. As funções de ligação consideradas são bastante flexíveis e robustas, contemplando funções de ligação simétricas como casos particulares. Devido a complexidade do modelo, nós discutimos a sua identificabilidade. A abordagem Bayesiana foi considerada e alguns algoritmos de Monte Carlo via Cadeia de Markov (MCMC) foram desenvolvidos. Nós descrevemos algumas ferramentas de seleção de modelos, os quais incluem o Critério de Informação da Deviance (DIC), a Pseudo-Verossimilhança Marginal e o Pseudo-Fator de Bayes. Adicionalmente, um estudo de simulação foi desenvolvido com dois objetivos; i) verificar a qualidade dos algoritmos desenvolvidos e ii) verificar a importância da escolha da função de ligação . No final da tese uma aplicação em um conjunto de dados real é considerada com o objetivo de ilustrar as metodologias e técnicas apresentadas. / Data sets with multivariate responses often appear in surveys where the data came from questionnaires. Opinion poll, sometimes simply referred to as a poll, are common examples of studies in which the responses are multivariate. One type poll that gain great prominence in Brazil in election years, is the survey of vote intent. However, despite the higher visibility of prognostic studies of election, opnion polls is a tool widely used to detect trends and positions of different social segments on various topics, be they political, social or governmental. We introduce in this work a class of multivariate regression models with asymmetric link functions to fit data sets with multivariate binary responses. The link functions here considered are quite flexible and robust, contemplating symmetrical link functions as special cases. Due to the complexity of the model, we discuss its identifiability. The Bayesian approach was considered and some Monte Carlo Markov Chain (MCMC) algorithms have been developed. Simulation studies have been developed with two objectives: i) verify the quality of the algorithms developed and ii) to verify the importance of choosing the link function. At the end of this work an application in a real data set is considered in order to illustrate the methodologies and techniques presented.

Conjuntos de dados com respostas multivariadas aparecem frequentemente em pesquisas em que os dados são provenientes de questionários. Exemplos mais comuns são pesquisas de opinião, mais especificamente, pesquisas de marketing em que a preferência do consumidor em potencial é avaliado: pelo produto, marca, preço, praça, promoção e etc. Um tipo pesquisa de opinião que ganha grande destaque no Brasil de dois em dois anos são as pesquisas eleitorais de intenção de votos. Nós introduzimos nesta tese uma classe de modelos de regressão multivariados com funções de ligação assimétricas para o ajuste de conjuntos de dados com respostas multivariadas binárias. As funções de ligação consideradas são bastante flexíveis e robustas, contemplando funções de ligação simétricas como casos particulares. Devido a complexidade do modelo, nós discutimos a sua identificabilidade. A abordagem Bayesiana foi considerada e alguns algoritmos de Monte Carlo via Cadeia de Markov (MCMC) foram desenvolvidos. Nós descrevemos algumas ferramentas de seleção de modelos, os quais incluem o Critério de Informação da Deviance (DIC), a Pseudo-Verossimilhança Marginal e o Pseudo-Fator de Bayes. Adicionalmente, um estudo de simulação foi desenvolvido com dois objetivos; i) verificar a qualidade dos algoritmos desenvolvidos e ii) verificar a importância da escolha da função de ligação . No final da tese uma aplicação em um conjunto de dados real é considerada com o objetivo de ilustrar as metodologias e técnicas apresentadas. / Data sets with multivariate responses often appear in surveys where the data came from questionnaires. Opinion poll, sometimes simply referred to as a poll, are common examples of studies in which the responses are multivariate. One type poll that gain great prominence in Brazil in election years, is the survey of vote intent. However, despite the higher visibility of prognostic studies of election, opnion polls is a tool widely used to detect trends and positions of different social segments on various topics, be they political, social or governmental. We introduce in this work a class of multivariate regression models with asymmetric link functions to fit data sets with multivariate binary responses. The link functions here considered are quite flexible and robust, contemplating symmetrical link functions as special cases. Due to the complexity of the model, we discuss its identifiability. The Bayesian approach was considered and some Monte Carlo Markov Chain (MCMC) algorithms have been developed. Simulation studies have been developed with two objectives: i) verify the quality of the algorithms developed and ii) to verify the importance of choosing the link function. At the end of this work an application in a real data set is considered in order to illustrate the methodologies and techniques presented.

This research explores factor analysis applied to data from skewed distributions for the general skew model, the selection-elliptical model, the selection-normal model, the skew-elliptical model and the skew-normal model for finite sample sizes. In terms of asymptotics, or large sample sizes, quasi-maximum likelihood methods are broached numerically. The skewed models are formed using selection distribution theory, which is based on Rao’s weighted distribution theory. The models assume the observed variable of the factor model is from a skewed distribution by defining the distribution of the unobserved common factors skewed and the unobserved unique factors symmetric. Numerical examples are provided using maximum likelihood selection skew-normal factor analysis. The numerical examples, such as maximum likelihood parameter estimation with the resolution of the “sign switching” problem and model fitting using likelihood methods, illustrate that the selection skew-normal factor analysis model better fits skew-normal data than does the normal factor analysis model.

This dissertation is composed of three articles describing novel approaches for analysis and modeling using multivariate skew-normal and skew-t distributions in econometrics and environmetrics. In the first article we introduce the Heckman selection-t model. Sample selection arises often as a result of the partial observability of the outcome of interest in a study. In the presence of sample selection, the observed data do not represent a random sample from the population, even after controlling for explanatory variables. Heckman introduced a sample-selection model to analyze such data and proposed a full maximum likelihood estimation method under the assumption of normality. The method was criticized in the literature because of its sensitivity to the normality assumption. In practice, data, such as income or expenditure data, often violate the normality assumption because of heavier tails. We first establish a new link between sample-selection models and recently studied families of extended skew-elliptical distributions. This then allows us to introduce a selection-t model, which models the error distribution using a Student’s t distribution. We study its properties and investigate the finite-sample performance of the maximum likelihood estimators for this model. We compare the performance of the selection-t model to the Heckman selection model and apply it to analyze ambulatory expenditures. In the second article we introduce a family of multivariate log-skew-elliptical distributions, extending the list of multivariate distributions with positive support. We investigate their probabilistic properties such as stochastic representations, marginal and conditional distributions, and existence of moments, as well as inferential properties. We demonstrate, for example, that as for the log-t distribution, the positive moments of the log-skew-t distribution do not exist. Our emphasis is on two special cases, the log-skew-normal and log-skew-t distributions, which we use to analyze U.S. precipitation data. Many commonly used statistical methods assume that data are normally distributed. This assumption is often violated in practice which prompted the development of more flexible distributions. In the third article we describe two such multivariate distributions, the skew-normal and the skew-t, and present commands for fitting univariate and multivariate skew-normal and skew-t regressions in the statistical software package Stata.

In the present work we study families of skew-probability distributions. We will gradually build concept of families of more and more general distributions. For us the most important ones are skew normal distribution, elliptical distri- bution and skew elliptical distribution. On the each of them we will present basic properties and visualize particular examples. At the end we will generate realizations of variates and propose how to estimate the original distribution.