# Sparsified Simultaneous Confidence Intervals for High-Dimensional Linear Models

arXiv: https://arxiv.org/abs/2307.07574 · HTML: https://mlsanity.com/paper/2307.07574/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 64 references that are in the map.
- First posted on arXiv 14 Jul 2023.

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

- [Exact post-selection inference, with application to the lasso](https://mlsanity.com/paper/1311.6238/) (2013)
- [Confidence Intervals and Hypothesis Testing for High-Dimensional Regression](https://mlsanity.com/paper/1306.3171/) (2013)
- [Exact Post-Selection Inference for Sequential Regression Procedures](https://mlsanity.com/paper/1401.3889/) (2014)
- [A significance test for the lasso](https://mlsanity.com/paper/1301.7161/) (2013)
- [Global and Simultaneous Hypothesis Testing for High-Dimensional Logistic Regression Models](https://mlsanity.com/paper/1805.06970/) (2018)
- [High-dimensional simultaneous inference with the bootstrap](https://mlsanity.com/paper/1606.03940/) (2016)
- [Model Selection for High Dimensional Quadratic Regression via Regularization](https://mlsanity.com/paper/1501.00049/) (2014)
- [Generic Error Bounds for the Generalized Lasso with Sub-Exponential Data](https://mlsanity.com/paper/2004.05361/) (2020)
- [A Bootstrap Lasso + Partial Ridge Method to Construct Confidence Intervals for Parameters in High-dimensional Sparse Linear Models](https://mlsanity.com/paper/1706.02150/) (2017)
- [Model Confidence Bounds for Variable Selection](https://mlsanity.com/paper/1611.09509/) (2016)
- [Model Selection Confidence Sets by Likelihood Ratio Testing](https://mlsanity.com/paper/1709.04342/) (2017)
- [Selection by Partitioning the Solution Paths](https://mlsanity.com/paper/1606.07358/) (2016)
- [Weak Signal Identification and Inference in Penalized Model Selection](https://mlsanity.com/paper/1611.04638/) (2016)

## Abstract (arXiv)

Statistical inference of the high-dimensional regression coefficients is challenging because the uncertainty introduced by the model selection procedure is hard to account for. A critical question remains unsettled; that is, is it possible and how to embed the inference of the model into the simultaneous inference of the coefficients? To this end, we propose a notion of simultaneous confidence intervals called the sparsified simultaneous confidence intervals. Our intervals are sparse in the sense that some of the intervals' upper and lower bounds are shrunken to zero (i.e., [0,0]), indicating the unimportance of the corresponding covariates. These covariates should be excluded from the final model. The rest of the intervals, either containing zero (e.g., [-1,1] or [0,1]) or not containing zero (e.g., [2,3]), indicate the plausible and significant covariates, respectively. The proposed method can be coupled with various selection procedures, making it ideal for comparing their uncertainty. For the proposed method, we establish desirable asymptotic properties, develop intuitive graphical tools for visualization, and justify its superior performance through simulation and real data analysis.