  Multivariate Gaussian regression is embedded into a general distributional
regression framework using flexible additive predictors determining all
distributional parameters. While this is relatively straightforward for the
means of the multivariate dependent variable, it is more challenging for the
full covariance matrix {\Sigma} due to two main difficulties: (i) ensuring
positive-definiteness of {\Sigma} and (ii) regularizing the high model
complexity. Both challenges are addressed by adopting a parameterization of
{\Sigma} based on its basic or modified Cholesky decomposition, respectively.
Unlike the decomposition into variances and a correlation matrix, the Cholesky
decomposition guarantees positive-definiteness for any predictor values
regardless of the distributional dimension. Thus, this enables linking all
distributional parameters to flexible predictors without any joint constraints
that would substantially complicate other parameterizations. Moreover, this
approach enables regularization of the flexible additive predictors through
penalized maximum likelihood or Bayesian estimation as for other distributional
regression models. Finally, the Cholesky decomposition allows to reduce the
number of parameters when the components of the multivariate dependent variable
have a natural order (typically time) and a maximum lag can be assumed for the
dependencies among the components.
