Quantile regression models are a powerful tool for studying different points of the conditional distribution of univariate response variables. Their multivariate counterpart extension though is not straightforward, starting with the definition of multivariate quantiles. We propose here a flexible Bayesian quantile regression model when the response variable is multivariate, where we are able to define a structured additive framework for all predictor variables. We build on previous ideas considering a directional approach to define the quantiles of a response variable with multiple outputs, and we define noncrossing quantiles in every directional quantile model. We define a Markov chain Monte Carlo (MCMC) procedure for model estimation, where the noncrossing property is obtained considering a Gaussian process design to model the correlation between several quantile regression models. We illustrate the results of these models using two datasets: one on dimensions of inequality in the population, such as income and health; the second on scores of students in the Brazilian High School National Exam, considering three dimensions for the response variable.

中文翻译：

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非交叉结构加性多输出贝叶斯分位数回归模型

分位数回归模型是研究单变量响应变量的条件分布的不同点的强大工具。从多元分位数的定义开始，它们的多元对应扩展并不是很简单。当响应变量是多变量时，我们在此提出了一种灵活的贝叶斯分位数回归模型，在该模型中，我们可以为所有预测变量定义结构化的加性框架。我们在先前的思想基础上考虑了使用方向性方法来定义具有多个输出的响应变量的分位数，并在每个方向分位数模型中定义了非交叉分位数。我们定义了一个马尔可夫链蒙特卡洛（MCMC）程序进行模型估计，其中考虑了高斯过程设计来建模几个分位数回归模型之间的相关性，从而获得非交叉属性。我们使用两个数据集来说明这些模型的结果：一个关于人口不平等的维度，例如收入和健康；第二个关于人口不平等的维度。考虑到响应变量的三个维度，第二次是巴西高中全国考试的学生分数。