# Two-component Mixture Model in the Presence of Covariates

arXiv: https://arxiv.org/abs/1810.07897 · HTML: https://mlsanity.com/paper/1810.07897/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Oct 2018.

## Built on by (most cited first, 4 of 4)

- [A Mean Field Approach to Empirical Bayes Estimation in High-dimensional Linear Regression](https://mlsanity.com/paper/2309.16843/) (2023)
- [On Parameter Estimation in Deviated Gaussian Mixture of Experts](https://mlsanity.com/paper/2402.05220/) (2024)
- [EM Approaches to Nonparametric Estimation for Mixture of Linear Regressions](https://mlsanity.com/paper/2510.14890/) (2025)
- [Empirical Bayes for correlated Gaussian sequence model](https://mlsanity.com/paper/2607.03596/) (2026)

## Abstract (arXiv)

In this paper, we study a generalization of the two-groups model in the presence of covariates --- a problem that has recently received much attention in the statistical literature due to its applicability in multiple hypotheses testing problems. The model we consider allows for infinite dimensional parameters and offers flexibility in modeling the dependence of the response on the covariates. We discuss the identifiability issues arising in this model and systematically study several estimation strategies. We propose a tuning parameter-free nonparametric maximum likelihood method, implementable via the EM algorithm, to estimate the unknown parameters. Further, we derive the rate of convergence of the proposed estimators --- in particular, we show that the finite sample Hellinger risk for every `approximate' nonparametric maximum likelihood estimator achieves a near-parametric rate (up to logarithmic multiplicative factors). In addition, we propose and theoretically study two `marginal' methods that are more scalable and easily implementable. We demonstrate the efficacy of our procedures through extensive simulation studies and relevant data analyses --- one arising from neuroscience and the other from astronomy. We also outline the application of our methods to multiple testing. The companion R package textttNPMLEmix implements all the procedures proposed in this paper.