# Nonparametric inference under shape constraints: past, present and future

arXiv: https://arxiv.org/abs/2509.26040 · HTML: https://mlsanity.com/paper/2509.26040/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 64 references that are in the map.
- First posted on arXiv 30 Sep 2025.

## Builds on (most cited first, 13 of 13)

- [How to Train Your Energy-Based Models](https://mlsanity.com/paper/2101.03288/) (2021)
- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [Isotonic regression with unknown permutations: Statistics, computation, and adaptation](https://mlsanity.com/paper/2009.02609/) (2020)
- [Optimality of Maximum Likelihood for Log-Concave Density Estimation and Bounded Convex Regression](https://mlsanity.com/paper/1903.05315/) (2019)
- [Optimal convex M-estimation via score matching](https://mlsanity.com/paper/2403.16688/) (2024)
- [Convex Regression in Multidimensions: Suboptimality of Least Squares Estimators](https://mlsanity.com/paper/2006.02044/) (2020)
- [A new computational framework for log-concave density estimation](https://mlsanity.com/paper/2105.11387/) (2021)
- [High-dimensional nonparametric density estimation via symmetry and shape constraints](https://mlsanity.com/paper/1903.06092/) (2019)
- [Log-concave Density Estimation with Independent Components](https://mlsanity.com/paper/2401.01500/) (2024)
- [Nonparametric, tuning-free estimation of S-shaped functions](https://mlsanity.com/paper/2107.07257/) (2021)
- [Optimal subgroup selection](https://mlsanity.com/paper/2109.01077/) (2021)
- [Isotonic subgroup selection](https://mlsanity.com/paper/2305.04852/) (2023)
- [Testing conditional independence under isotonicity](https://mlsanity.com/paper/2501.06133/) (2025)

## Built on by (most cited first, 3 of 3)

- [Density estimation for Hellinger via minimum-distance estimators: mixtures of Gaussians, log-concave, and more](https://mlsanity.com/paper/2606.11469/) (2026)
- [Calibration without labels in multiple testing](https://mlsanity.com/paper/2606.19737/) (2026)
- [Instance-Optimal Adaptive Location Estimation via Multiscale Mid-Summaries](https://mlsanity.com/paper/2609.20749/) (2026)

## Abstract (arXiv)

We survey the field of nonparametric inference under shape constraints, providing a historical overview and a perspective on its current state. An outlook and some open problems offer thoughts on future directions.