Generalized additive models for location, scale and shape (GAMLSS) are a popular extension to mean regression models where each parameter of an arbitrary distribution is modelled through covariates. While such models have been developed for univariate and bivariate responses, the truly multivariate case remains extremely challenging for both computational and theoretical reasons. Alternative approaches to GAMLSS may allow for higher dimensional response vectors to be modelled jointly but often assume a fixed dependence structure not depending on covariates or are limited with respect to modelling flexibility or computational aspects. We contribute to this gap in the literature and propose a truly multivariate distributional model, which allows one to benefit from the flexibility of GAMLSS even when the response has dimension larger than two or three. Building on copula regression, we model the dependence structure of the response through a Gaussian copula, while the marginal distributions can vary across components. Our model is highly parameterized but estimation becomes feasible with Bayesian inference employing shrinkage priors. We demonstrate the competitiveness of our approach in a simulation study and illustrate how it complements existing models along the examples of childhood malnutrition and a yet unexplored data set on traffic detection in Berlin.

翻译：位置、尺度与形状的广义可加模型（GAMLSS）是均值回归模型的一种流行扩展，其中任意分布的每个参数均可通过协变量建模。虽然此类模型已针对单变量和双变量响应变量开发，但真正的多元情形因计算和理论原因仍极具挑战性。GAMLSS的替代方法或许允许对更高维响应向量进行联合建模，但通常假设固定的依赖结构不依赖于协变量，或在建模灵活性或计算层面存在局限。本文针对这一文献空白作出贡献，提出一种真正的多元分布模型，即使当响应变量维度大于二或三维时，仍能受益于GAMLSS的灵活性。基于Copula回归，我们通过高斯Copula对响应变量的依赖结构建模，而各分量的边缘分布可互不相同。我们的模型具有高参数化特性，但采用收缩先验的贝叶斯推断使参数估计变得可行。我们通过模拟研究证明了所提方法的竞争力，并以儿童营养不良案例和柏林交通检测这一尚未被探索的数据集为例，阐释了该方法如何对现有模型形成补充。