The statistical analysis of univariate quantiles is a well developed research topic. However, there is a need for research in multivariate quantiles. We construct bivariate (conditional) quantiles using the level curves of vine copula based bivariate regression model. Vine copulas are graph theoretical models identified by a sequence of linked trees, which allow for separate modelling of marginal distributions and the dependence structure. We introduce a novel graph structure model (given by a tree sequence) specifically designed for a symmetric treatment of two responses in a predictive regression setting. We establish computational tractability of the model and a straight forward way of obtaining different conditional distributions. Using vine copulas the typical shortfalls of regression, as the need for transformations or interactions of predictors, collinearity or quantile crossings are avoided. We illustrate the copula based bivariate level curves for different copula distributions and show how they can be adjusted to form valid quantile curves. We apply our approach to weather measurements from Seoul, Korea. This data example emphasizes the benefits of the joint bivariate response modelling in contrast to two separate univariate regressions or by assuming conditional independence, for bivariate response data set in the presence of conditional dependence.

翻译：单变量分位数的统计分析是一个成熟的研究领域，然而多变量分位数研究仍存在需求。本文利用基于藤Copula双变量回归模型的水平曲线构建了双变量（条件）分位数。藤Copula是由一系列连接树构成的图论模型，能够分别对边缘分布和相依结构进行建模。我们提出了一种专为预测回归场景中对称处理两个响应变量而设计的新型图结构模型（由树序列表示）。我们证明了该模型的计算可行性，并给出了获取不同条件分布的直观方法。通过使用藤Copula，避免了回归中常见的典型缺陷（如预测变量的变换或交互作用需求、共线性或分位数交叉问题）。我们展示了不同Copula分布下基于Copula的双变量水平曲线，并阐述了如何将其调整为有效的分位数曲线。将所提方法应用于韩国首尔的气象测量数据。该数据示例凸显了在存在条件相依性的双变量响应数据集中，联合双变量响应建模相较于两个独立单变量回归或假设条件独立性的优势。