Understand
We consider the problem of variable selection in high-dimensional statistical models where the goal is to report a set of variables, out of many predictors $X_1, \dotsc, X_p$, that are relevant to a response of interest.
- For linear high-dimensional model, where the number of parameters exceeds the number of samples $(p>n)$, we propose a procedure for variables selection and prove that it controls the "directional" false discovery rate (FDR) below a pre-assigned significance level $q\in [0,1]$.
- We further analyze the statistical power of our framework and show that for designs with subgaussian rows and a common precision matrix $\Omega\in\mathbb{R}^{p\times p}$, if the minimum nonzero parameter $\theta_{\min}$ satisfies $$\sqrt{n} \theta_{\min} - \sigma \sqrt{2(\max_{i\in [p]}\Omega_{ii})\log\left(\frac{2p}{qs_0}\right)} \to \infty\,,$$ then this procedure achieves asymptotic power one.
- Our framework is built upon the debiasing approach and assumes the standard condition $s_0 = o(\sqrt{n}/(\log p)^2)$, where $s_0$ indicates the number of true positives among the $p$ features.