# Optimal shrinkage estimation in heteroscedastic hierarchical linear models

arXiv: https://arxiv.org/abs/1503.06262 · HTML: https://mlsanity.com/paper/1503.06262/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Mar 2015.

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

- [Structured Dynamic Pricing: Optimal Regret in a Global Shrinkage Model](https://mlsanity.com/paper/2303.15652/) (2023)
- [Empirical Bayes Estimation with Side Information: A Nonparametric Integrative Tweedie Approach](https://mlsanity.com/paper/2308.05883/) (2023)
- [PEBS: Per-rater Empirical-Bayes Shrinkage for RLHF Reward-Model Calibration](https://mlsanity.com/paper/2606.27578/) (2026)

## Abstract (arXiv)

Shrinkage estimators have profound impacts in statistics and in scientific and engineering applications. In this article, we consider shrinkage estimation in the presence of linear predictors. We formulate two heteroscedastic hierarchical regression models and study optimal shrinkage estimators in each model. A class of shrinkage estimators, both parametric and semiparametric, based on unbiased risk estimate (URE) is proposed and is shown to be (asymptotically) optimal under mean squared error loss in each model. Simulation study is conducted to compare the performance of the proposed methods with existing shrinkage estimators. We also apply the method to real data and obtain encouraging and interesting results.