# Post-selection point and interval estimation of signal sizes in Gaussian samples

arXiv: https://arxiv.org/abs/1405.3340 · HTML: https://mlsanity.com/paper/1405.3340/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 May 2014.

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

- [Confidence on the Focal: Conformal Prediction with Selection-Conditional Coverage](https://mlsanity.com/paper/2403.03868/) (2024)
- [Adaptive Noisy Data Augmentation for Regularized Estimation and Inference in Generalized Linear Models](https://mlsanity.com/paper/2204.08574/) (2022)
- [Online Selective Conformal Prediction with Asymmetric Rules: A Permutation Test Approach](https://mlsanity.com/paper/2602.10018/) (2026)

## Abstract (arXiv)

We tackle the problem of the estimation of a vector of means from a single vector-valued observation y. Whereas previous work reduces the size of the estimates for the largest (absolute) sample elements via shrinkage (like James-Stein) or biases estimated via empirical Bayes methodology, we take a novel approach. We adapt recent developments by Lee et al (2013) in post selection inference for the Lasso to the orthogonal setting, where sample elements have different underlying signal sizes. This is exactly the setup encountered when estimating many means. It is shown that other selection procedures, like selecting the K largest (absolute) sample elements and the Benjamini-Hochberg procedure, can be cast into their framework, allowing us to leverage their results. Point and interval estimates for signal sizes are proposed. These seem to perform quite well against competitors, both recent and more tenured. Furthermore, we prove an upper bound to the worst case risk of our estimator, when combined with the Benjamini-Hochberg procedure, and show that it is within a constant multiple of the minimax risk over a rich set of parameter spaces meant to evoke sparsity.