A unified theory of confidence intervals for high-dimensional precision matrix

Abstract

Precision matrix inference is of fundamental importance nowadays in high-dimensional data analysis for measuring conditional dependence. Despite the fast growing literature, developing approaches to make simultaneous inference for precision matrix with low computational cost is still in urgent need. In this paper, we apply bootstrap-assisted procedure to conduct simultaneous inference for high-dimensional precision matrix based on the recent de-biased nodewise Lasso estimator, which does not require the irrepresentability condition and is easy to implement with low computational cost. Furthermore, we summary a unified framework to perform simultaneous confidence intervals for high-dimensional precision matrix under the sub-Gaussian case. We show that as long as some precision matrix estimation effects are satisfied, our procedure can focus on different precision matrix estimation methods which owns great flexibility. Besides, distinct from earlier Bonferroni-Holm procedure, this bootstrap method is asymptotically nonconservative. Both numerical results confirm the theoretical results and computational advantage of our method.