# A Nonparametric Maximum Likelihood Approach to Mixture of Regression

arXiv: https://arxiv.org/abs/2108.09816 · HTML: https://mlsanity.com/paper/2108.09816/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Aug 2021.

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

- [Imbalanced Mixed Linear Regression](https://mlsanity.com/paper/2301.12559/) (2023)
- [Neural-g: A Deep Learning Framework for Mixing Density Estimation](https://mlsanity.com/paper/2406.05986/) (2024)
- [EM Approaches to Nonparametric Estimation for Mixture of Linear Regressions](https://mlsanity.com/paper/2510.14890/) (2025)

## Abstract (arXiv)

We study mixture of linear regression (random coefficient) models, which capture population heterogeneity by allowing the regression coefficients to follow an unknown distribution G^*. In contrast to common parametric methods that fix the mixing distribution form and rely on the EM algorithm, we develop a fully nonparametric maximum likelihood estimator (NPMLE). We show that this estimator exists under broad conditions and can be computed via a discrete approximation procedure inspired by the exemplar method. We further establish theoretical guarantees demonstrating that the NPMLE achieves near-parametric rates in estimating the conditional density of Y|X, both for fixed and random designs, when σ is known and G^* has compact support. In the random design setting, we also prove consistency of the estimated mixing distribution in the L'evy-Prokhorov distance. Numerical experiments indicate that our approach performs well and additionally enables posterior-based individualized coefficient inference through an empirical Bayes framework.