# Distributional (Single) Index Models

arXiv: https://arxiv.org/abs/2006.09219 · HTML: https://mlsanity.com/paper/2006.09219/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jun 2020.

## Built on by (most cited first, 6 of 6)

- [Machine learning methods for postprocessing ensemble forecasts of wind gusts: A systematic comparison](https://mlsanity.com/paper/2106.09512/) (2021)
- [A review of predictive uncertainty estimation with machine learning](https://mlsanity.com/paper/2209.08307/) (2022)
- [Isotonic Recalibration under a Low Signal-to-Noise Ratio](https://mlsanity.com/paper/2301.02692/) (2023)
- [Generative and Nonparametric Approaches for Conditional Distribution Estimation: Methods, Perspectives, and Comparative Evaluations](https://mlsanity.com/paper/2601.22650/) (2026)
- [Invariant Probabilistic Prediction](https://mlsanity.com/paper/2309.10083/) (2023)
- [Isotonic Survival Regression: Calibrated Survival Distributions from Deep Cox Models](https://mlsanity.com/paper/2605.16571/) (2026)

## Abstract (arXiv)

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.