  A Distributional (Single) Index Model (DIM) is a semi-parametric model for
distributional regression, that is, estimation of conditional distributions
given covariates. The method is a combination of classical single index models
for the estimation of the conditional mean of a response given covariates, and
isotonic distributional regression. The model for the index is parametric,
whereas the conditional distributions are estimated non-parametrically under a
stochastic ordering constraint. We show consistency of our estimators and apply
them to a highly challenging data set on the length of stay (LoS) of patients
in intensive care units. We use the model to provide skillful and calibrated
probabilistic predictions for the LoS of individual patients, that outperform
the available methods in the literature.
