Quantile regression (QR) (Koenker and Bassett, 1978), as a comprehensive extension to standard mean regression, has been steadily promoted from both theoretical and applied aspects. Bayesian quantile regression (BQR), which deals with unknown parameter estimation and model uncertainty, is a newly proposed tool of QR. This thesis aims to make some novel contributions to the following three issues related to QR. First, whereas QR for continuous responses has received much attention in literatures, QR for discrete responses has received far less attention. Second, conventional QR methods often show that QR curves crossing lead to invalid distributions for the response. In particular, given a set of covariates, it may turn out, for example, that the predicted 95th percentile of the response is smaller than the 90th percentile for some values of the covariates. Third, mean-based clustering methods are widely developed, but need improvements to deal with clustering extreme-type, heavy tailed-type or outliers problems. This thesis focuses on methods developed over these three challenges: modelling quantile regression with discrete responses, ensuring non-crossing quantile curves for any given sample and modelling tails for collinear data with outliers. The main contributions are listed as below: * The first challenge is studied in Chapter 2, in which a general method for Bayesian inference of regression models beyond the mean with discrete responses is developed. In particular, this method is developed for both Bayesian quantile regression and Bayesian expectile regression. This method provides a direct Bayesian approach to these regression models with a simple and intuitive interpretation of the regression results. The posterior distribution under this approach is shown to not only be coherent to the response variable, irrespective of its true distribution, but also proper in relation to improper priors for unknown model parameters. * Chapter 3 investigates a new kernel-weighted likelihood smoothing quantile regression method. The likelihood is based on a normal scale-mixture representation of an asymmetric Laplace distribution (ALD). This approach benefits of the same good design adaptation just as the local quantile regression (Spokoiny et al., 2014) does and ensures non-crossing quantile curves for any given sample. * In Chapter 4, we introduce an asymmetric Laplace distribution to model the response variable using profile regression, a Bayesian non-parametric model for clustering responses and covariates simultaneously. This development allows us to model more accurately for clusters which are asymmetric and predict more accurately for extreme values of the response variable and/or outliers. In addition to the three major aforementioned challenges, this thesis also addresses other important issues such as smoothing extreme quantile curves and avoiding insensitive to heteroscedastic errors as well as outliers in the response variable. The performances of all the three developments are evaluated via both simulation studies and real data analysis.

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This thesis explores the problems of fitting antedependence (AD) models and partial antecorrelation (PAC) models to continuous non-Gaussian longitudinal data. AD models impose certain conditional independence relations among the measurements within each subject, while PAC models characterize the partial correlation relations. The models are parsimonious and useful for data exhibiting time-dependent correlations. Since the relation of conditional independence among variables is rather restrictive, we first consider an autoregressively characterized PAC model with independent asymmetric Laplace (ALD) innovations and prove that this model is an AD model. The ALD distribution previously has been applied to quantile regression and has shown promise for modeling asymmetrically distributed ecological data. In addition, the double exponential distribution, a special case of the ALD, has played an important role in fitting symmetric finance and hydrology data. We give the distribution of a linear combination of independent standard ALD variables in order to derive marginal distributions for the model. For the model estimation problem, we propose an iterative algorithm for the maximum likelihood estimation. The estimation accuracy is illustrated by some numerical examples as well as some longitudinal data sets. The second component of this dissertation focuses on AD multivariate skew normal models. The multivariate skew normal distribution not only shares some nice properties with multivariate normal distributions but also allows for any value of skewness. We derive necessary and sufficient conditions on the shape and covariance parameters for multivariate skew normal variables to be AD(p) for some p. Likelihood-based estimation for balanced and monotone missing data as well as likelihood ratio hypothesis tests for the order of antedependence and for zero skewness under the models are presented. Since the class of skew normal random variables is closed under the addition of independent standard normal random variables, we then consider an autoregressively characterized PAC model with a combination of independent skew normal and normal innovations. Explicit expressions for the marginals, which all have skew normal distributions, and maximum likelihood estimates of model parameters, are given. Numerical results show that these three proposed models may provide reasonable fits to some continuous non-Gaussian longitudinal data sets. Furthermore, we compare the fits of these models to the Treatment A cattle growth data using penalized likelihood criteria, and demonstrate that the AD(2) multivariate skew normal model fits the data best among those proposed models.

Heteroscedasticity (or time-dependent volatility) in economic and financial time series has been recognized for decades. Still, heteroscedasticity is surprisingly often neglected by practitioners and researchers. This may lead to inefficient procedures. Much of the work in this thesis is about finding more effective ways to deal with heteroscedasticity in economic and financial data. Paper I suggest a filter that, unlike the Box-Cox transformation, does not assume that the heteroscedasticity is a power of the expected level of the series. This is achieved by dividing the time series by a moving average of its standard deviations smoothed by a Hodrick-Prescott filter. It is shown that the filter does not colour white noise. An appropriate removal of heteroscedasticity allows more effective analyses of heteroscedastic time series. A few examples are presented in Paper II, III and IV of this thesis. Removing the heteroscedasticity using the proposed filter enables efficient estimation of the underlying probability distribution of economic growth. It is shown that the mixed Normal - Asymmetric Laplace (NAL) distributional fit is superior to the alternatives. This distribution represents a Schumpeterian model of growth, the driving mechanism of which is Poisson (Aghion and Howitt, 1992) distributed innovations. This distribution is flexible and has not been used before in this context. Another way of circumventing strong heteroscedasticity in the Dow Jones stock index is to divide the data into volatility groups using the procedure described in Paper III. For each such group, the most accurate probability distribution is searched for and is used in density forecasting. Interestingly, the NAL distribution fits best also here. This could hint at a new analogy between the financial sphere and the real economy, further investigated in Paper IV. These series are typically heteroscedastic, making standard detrending procedures, such as Hodrick-Prescott or Baxter-King, inadequate. Prior to this comovement study, the univariate and bivariate frequency domain results from these filters are compared to the filter proposed in Paper I. The effect of often neglected heteroscedasticity may thus be studied.

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.

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.