# Directional FDR Control for Sub-Gaussian Sparse GLMs

arXiv: https://arxiv.org/abs/2105.00393 · HTML: https://mlsanity.com/paper/2105.00393/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 41 references that are in the map.
- First posted on arXiv 2 May 2021.

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

- [Panning for Gold: Model-X Knockoffs for High-dimensional Controlled Variable Selection](https://mlsanity.com/paper/1610.02351/) (2016)
- [Controlling the false discovery rate via knockoffs](https://mlsanity.com/paper/1404.5609/) (2014)
- [Concentration Inequalities for Statistical Inference](https://mlsanity.com/paper/2011.02258/) (2020)
- [Global and Simultaneous Hypothesis Testing for High-Dimensional Logistic Regression Models](https://mlsanity.com/paper/1805.06970/) (2018)
- [Gaussian Graphical Model Estimation with False Discovery Rate Control](https://mlsanity.com/paper/1306.0976/) (2013)
- [Elastic-net Regularized High-dimensional Negative Binomial Regression: Consistency and Weak Signals Detection](https://mlsanity.com/paper/1712.03412/) (2017)
- [False Discovery Rate Control via Debiased Lasso](https://mlsanity.com/paper/1803.04464/) (2018)
- [RANK: Large-Scale Inference with Graphical Nonlinear Knockoffs](https://mlsanity.com/paper/1709.00092/) (2017)
- [Phase Transition and Regularized Bootstrap in Large Scale t-tests with False Discovery Rate Control](https://mlsanity.com/paper/1310.4371/) (2013)
- [Confidence intervals for high-dimensional Cox models](https://mlsanity.com/paper/1803.01150/) (2018)
- [A Revisit to De-biased Lasso for Generalized Linear Models](https://mlsanity.com/paper/2006.12778/) (2020)
- [An error bound for Lasso and Group Lasso in high dimensions](https://mlsanity.com/paper/1912.11398/) (2019)
- [Weighted Lasso Estimates for Sparse Logistic Regression: Non-asymptotic Properties with Measurement Error](https://mlsanity.com/paper/2006.06136/) (2020)

## Built on by (most cited first, 2 of 2)

- [Heterogeneous Overdispersed Count Data Regressions via Double Penalized Estimations](https://mlsanity.com/paper/2110.03552/) (2021)
- [High-dimensional Inference and FDR Control for Simulated Markov Random Fields](https://mlsanity.com/paper/2202.05612/) (2022)

## Abstract (arXiv)

High-dimensional sparse generalized linear models (GLMs) have emerged in the setting that the number of samples and the dimension of variables are large, and even the dimension of variables grows faster than the number of samples. False discovery rate (FDR) control aims to identify some small number of statistically significantly nonzero results after getting the sparse penalized estimation of GLMs. Using the CLIME method for precision matrix estimations, we construct the debiased-Lasso estimator and prove the asymptotical normality by minimax-rate oracle inequalities for sparse GLMs. In practice, it is often needed to accurately judge each regression coefficient's positivity and negativity, which determines whether the predictor variable is positively or negatively related to the response variable conditionally on the rest variables. Using the debiased estimator, we establish multiple testing procedures. Under mild conditions, we show that the proposed debiased statistics can asymptotically control the directional (sign) FDR and directional false discovery variables at a pre-specified significance level. Moreover, it can be shown that our multiple testing procedure can approximately achieve a statistical power of 1. We also extend our methods to the two-sample problems and propose the two-sample test statistics. Under suitable conditions, we can asymptotically achieve directional FDR control and directional FDV control at the specified significance level for two-sample problems. Some numerical simulations have successfully verified the FDR control effects of our proposed testing procedures, which sometimes outperforms the classical knockoff method.