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Interestingly, the corresponding limiting distribution is the chi-square or normal, depending on if the co-dimension of a test is finite or increases with the sample size, leading to asymptotic similar tests. This goes beyond the classical Wilks phenomenon. Numerically, we demonstrate that the proposed method performs well against it competitors in various scenarios. Finally, we apply the proposed method to infer linkages in brain network analysis based on MRI data, to contrast Alzheimer's disease patients against healthy subjects. Supp