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Bayesian quantile regression using the skew exponential power distribution AYDA TAJDINI a taj@math iut ac ir January 25 2020 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Zahra Saberi z saberi@cc iut ac irAdvisor Dr Rehraneh Rikhtehgaran r rikhtehgaran@cc iut ac ir2000 MSC 62E99 62F15 62J99 Keywords Bayesian quantile regression Skew exponential power distribution Asymmetric Laplace distribution Variable selection Ridge regression Lasso regression Adaptive lasso regression Markov Monte Carlo chain meth ods Abstract This M Sc thesis is based on the following papers Bayesian quantile regression using the skew exponen M B M B L P tial power distribution Computational Statistics and Data Analysis 126 2018 92 111 L Bayesian Regularized Quantile Regression Bayesian Analysis Number 1 Q L R X N 2004 1 26 Statistics is one of the tools used in many sciences for data analysis One of the most commonly used statisticalmethods especially in economics social sciences etc is the regression model Mean regression expresses the rela tionship between the conditional mean of a response variable in terms of one or more independent variables However sometimes this method will perform poorly in data analysis For example in cases where error distributionis not normal or if variance heterogeneity exists least squares estimators are sensitive to outliers and lead to biasedestimators On the other hand least squares regression expresses the relationship between covariates and mean re sponse variable while in many cases the goal is to find the relationship between independent variables with wholeparts of the conditional distribution of the response variable such as quantiles In these cases the quantile regression method can be used In order to easily implement bayesian methods to obtainparameter estimation first a distribution for the response variable and prior distribution for the parameters are con sidered then the Markov Monte Carlo chain methods are used to generate the samples from the posterior distribution Although the asymmetric laplace distribution is widely used in the bayesian quantile regression model but whenthe observations include outliers and the distribution of observations is heavy tailed this distribution will not workproperly Also for each specific quantile such as 0 the skewness of this distribution is constant In recent years researchers have attempted to introduce a more flexible skew distribution that is used to estimatethe parameters of a bayesian quantile regression model based on outliers and heavy tailed observations So in thisthesis we propose a bayesian quantile regression model with skewed exponential power distribution error to solvethis problem First by applying the penalized least squares regression methods we estimate the regression coefficients and thevariable selection process using the lasso and adaptive lasso based on the asymmetric laplace distribution as the er ror distribution These methods reduce the dimension of the model parameters So bayesian inference and samplegeneration from the posterior distribution are made using Gibbs sampling algorithm In the following we introducethe skew exponential power distribution and describe the bayesian quantile regression model using it as the error dis tribution In this distribution family posterior distribution sampling is performed using the independent Metropolis Hastings within Gibbs sampling algorithm which will also be associated with the variable selection process Thenwe use simulation and applied studies to evaluate the performance of the proposed model

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