# Selective inference with unknown variance via the square-root LASSO

arXiv: https://arxiv.org/abs/1504.08031 · HTML: https://mlsanity.com/paper/1504.08031/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Apr 2015.

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

- [Tree-Values: selective inference for regression trees](https://mlsanity.com/paper/2106.07816/) (2021)
- [Black-box Selective Inference via Bootstrapping](https://mlsanity.com/paper/2203.14504/) (2022)
- [Inferring independent sets of Gaussian variables after thresholding correlations](https://mlsanity.com/paper/2211.01521/) (2022)
- [Enhancing Credit Risk Prediction: A Multi-stage Ensemble Pipeline](https://mlsanity.com/paper/2509.22381/) (2025)

## Abstract (arXiv)

There has been much recent work on inference after model selection when the noise level is known, however, σ is rarely known in practice and its estimation is difficult in high-dimensional settings. In this work we propose using the square-root LASSO (also known as the scaled LASSO) to perform selective inference for the coefficients and the noise level simultaneously. The square-root LASSO has the property that choosing a reasonable tuning parameter is scale-free, namely it does not depend on the noise level in the data. We provide valid p-values and confidence intervals for the coefficients after selection, and estimates for model specific variance. Our estimates perform better than other estimates of σ2 in simulation.