Distributional (Single) Index Models

Abstract

A Distributional (Single) Index Model (DIM) is a semiparametric 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 nonparametrically under a stochastic ordering constraint. We show consistency of our estimators and apply them to a highly challenging dataset 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, which outperform the available methods in the literature.

Keywords:

• Distributional regression
• Intensive care unit length of stay
• Probabilistic forecast
• Single-index model
• Stochastic ordering constraint

Supplementary Materials

The supplementary material contains further details and results for the case study.

Acknolwedgments

The authors thank to the Swiss Society of Intensive Care Medicine for providing the data. The work has greatly benefitted from discussions with Lutz Dümbgen, Tilmann Gneiting, Alexander Jordan, and Alexandre Mösching.

Additional information

Funding

A. Henzi and J. F. Ziegel gratefully acknowledge financial support from the Swiss National Science Foundation.