  Prediction models often fail if train and test data do not stem from the same
distribution. Out-of-distribution (OOD) generalization to unseen, perturbed
test data is a desirable but difficult-to-achieve property for prediction
models and in general requires strong assumptions on the data generating
process (DGP). In a causally inspired perspective on OOD generalization, the
test data arise from a specific class of interventions on exogenous random
variables of the DGP, called anchors. Anchor regression models, introduced by
Rothenh\"ausler et al. (2018), protect against distributional shifts in the
test data by employing causal regularization. However, so far anchor regression
has only been used with a squared-error loss which is inapplicable to common
responses such as censored continuous or ordinal data. Here, we propose a
distributional version of anchor regression which generalizes the method to
potentially censored responses with at least an ordered sample space. To this
end, we combine a flexible class of parametric transformation models for
distributional regression with an appropriate causal regularizer under a more
general notion of residuals. In an exemplary application and several simulation
scenarios we demonstrate the extent to which OOD generalization is possible.
