Minimax detection boundary and sharp optimal test for Gaussian graphical models

Abstract In this article, we derive the minimax detection boundary for testing a sub-block of variables in a precision matrix under the Gaussian distribution. Compared to the results on the minimum rate of signals for testing precision matrices in literature, our result gives the exact minimum signal strength in a precision matrix that can be detected. We propose a thresholding test that is able to achieve the minimax detection boundary under certain cases by adaptively choosing the threshold level. The asymptotic distribution of the thresholding statistic for precision matrices is derived. Power analysis is conducted to show the proposed test is powerful against sparse and weak signals, which cannot be detected by the existing Lmax and L2 tests. Simulation studies show the proposed test has an accurate size around the nominal level and is more powerful than the existing tests for detecting sparse and weak signals in precision matrices. Real data analysis on brain imaging data is carried out to illustrate the utility of the proposed test in practice, which reveals functional connectivity between brain regions for Alzheimer’s disease patients and normal healthy people.