# A Bootstrap Lasso + Partial Ridge Method to Construct Confidence Intervals for Parameters in High-dimensional Sparse Linear Models

arXiv: https://arxiv.org/abs/1706.02150 · HTML: https://mlsanity.com/paper/1706.02150/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jun 2017.

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

- [Interval Estimation of Coefficients in Penalized Regression Models of Insurance Data](https://mlsanity.com/paper/2410.01008/) (2024)
- [Two-Stage Robust and Sparse Distributed Statistical Inference for Large-Scale Data](https://mlsanity.com/paper/2208.08230/) (2022)
- [Robust Inference for High-Dimensional Linear Models via Residual Randomization](https://mlsanity.com/paper/2106.07717/) (2021)
- [Sparsified Simultaneous Confidence Intervals for High-Dimensional Linear Models](https://mlsanity.com/paper/2307.07574/) (2023)

## Abstract (arXiv)

Constructing confidence intervals for the coefficients of high-dimensional sparse linear models remains a challenge, mainly because of the complicated limiting distributions of the widely used estimators, such as the lasso. Several methods have been developed for constructing such intervals. Bootstrap lasso+ols is notable for its technical simplicity, good interpretability, and performance that is comparable with that of other more complicated methods. However, bootstrap lasso+ols depends on the beta-min assumption, a theoretic criterion that is often violated in practice. Thus, we introduce a new method, called bootstrap lasso+partial ridge, to relax this assumption. Lasso+partial ridge is a two-stage estimator. First, the lasso is used to select features. Then, the partial ridge is used to refit the coefficients. Simulation results show that bootstrap lasso+partial ridge outperforms bootstrap lasso+ols when there exist small, but nonzero coefficients, a common situation that violates the beta-min assumption. For such coefficients, the confidence intervals constructed using bootstrap lasso+partial ridge have, on average, 50% larger coverage probabilities than those of bootstrap lasso+ols. Bootstrap lasso+partial ridge also has, on average, 35% shorter confidence interval lengths than those of the de-sparsified lasso methods, regardless of whether the linear models are misspecified. Additionally, we provide theoretical guarantees for bootstrap lasso+partial ridge under appropriate conditions, and implement it in the R package "HDCI."