TABLE 5.

Average power and type-I error rate for the hypotheses $H_{0,j}^{\mathrm{*}}:{\beta }_j^{\mathrm{*}}=0$ for $j=1,\dots ,p$, as defined in (5.1)—(5.3), under the scale-free graph setting with $p=500$. Results are shown for various values of $\rho ,n$, SNR. Methods for comparison include LDPE, SSLasso, dScore, and the naïve score test with tuning parameter ${\lambda }_{min}$, ${\lambda }_{\mathit{1}SE}$ and ${\lambda }_{sup.}$. The error variance is estimated either through the scaled lasso (Sun and Zhang, 2012) (S-Z) or Theorem 2.2 (T2.2). Note that as mentioned in Section 3, we do not combine ${\lambda }_{sup}$ with T2.2

	$\rho$ $n$ SNR	0.1	100 0.3	0.5	0.1	0.2 200 0.3	0.5	0.1	400 0.3	0.5
	LDPE ${\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ S-Z	0.400	0.773	0.910	0.627	0.973	1.000	0.923	1.000	1.000
	SSLasso ${\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ S-Z	0.410	0.770	0.950	0.650	0.970	1.000	0.910	1.000	1.000
Powerstrong	dScore ${\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ S-Z	0.330	0.643	0.857	0.547	0.957	1.000	0.887	1.000	1.000
	nScore ${\lambda }_{\text{sup}}$ S-Z	0.403	0.847	0.960	0.727	0.990	0.997	0.940	1.000	1.000
	nScore ${\lambda }_{\text{1SE}}$ S-Z	0.427	0.763	0.893	0.677	0.977	1.000	0.957	1.000	1.000
	nScore ${\lambda }_{1\mathrm{S}\mathrm{E}}$ T2.2	0.357	0.763	0.890	0.680	0.977	1.000	0.910	1.000	1.000
	LDPE ${\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ S-Z	0.064	0.083	0.056	0.054	0.059	0.079	0.070	0.079	0.113
	SSLasso ${\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ S-Z	0.081	0.087	0.060	0.066	0.061	0.086	0.069	0.086	0.113
Powerweak	dScore ${\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ S-Z	0.044	0.056	0.036	0.039	0.039	0.060	0.046	0.056	0.093
	nScore ${\lambda }_{\text{sup}}$ S-Z	0.061	0.091	0.109	0.070	0.109	0.107	0.097	0.103	0.114
	nScore ${\lambda }_{\text{1SE}}$ S-Z	0.080	0.077	0.059	0.060	0.061	0.061	0.083	0.076	0.101
	nScore ${\lambda }_{1\mathrm{S}\mathrm{E}}$ T2.2	0.067	0.070	0.070	0.054	0.071	0.069	0.079	0.094	0.123
	LDPE ${\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ S-Z	0.051	0.052	0.051	0.049	0.051	0.047	0.050	0.051	0.049
	SSLasso ${\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ S-Z	0.056	0.056	0.056	0.054	0.055	0.053	0.053	0.054	0.054
T1 Error	dScore ${\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ S-Z	0.035	0.040	0.040	0.033	0.036	0.034	0.035	0.037	0.034
	nScore ${\lambda }_{\text{sup}}$ S-Z	0.069	0.082	0.095	0.064	0.083	0.079	0.065	0.068	0.050
	nScore ${\lambda }_{1\mathrm{S}\mathrm{E}}$ S-Z	0.061	0.057	0.048	0.056	0.055	0.040	0.060	0.046	0.046
	nScore ${\lambda }_{\text{1SE}}$ T2.2	0.049	0.052	0.049	0.054	0.053	0.050	0.056	0.049	0.050
	$\rho$ $n$ SNR	0.1	100 0.3	0.5	0.1	0.6 200 0.3	0.5	0.1	400 0.3	0.5
	LDPE ${\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ S-Z	0.330	0.783	0.947	0.627	0.980	1.000	0.887	1.000	1.000
	SSLasso ${\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ S-Z	0.347	0.790	0.957	0.623	0.987	1.000	0.867	1.000	1.000
Powerstrong	dScore ${\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ S-Z	0.270	0.673	0.883	0.533	0.960	0.993	0.863	1.000	1.000
	nScore ${\lambda }_{\text{sup}}$ S-Z	0.430	0.790	0.933	0.707	0.977	1.000	0.923	1.000	1.000
	nScore ${\lambda }_{1\mathrm{S}\mathrm{E}}$ S-Z	0.357	0.767	0.887	0.677	0.980	0.997	0.937	1.000	1.000
	nScore ${\lambda }_{\text{1SE}}$ T2.2	0.340	0.697	0.907	0.637	0.973	1.000	0.950	1.000	1.000
	LDPE ${\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ S-Z	0.031	0.046	0.063	0.064	0.074	0.076	0.054	0.077	0.119
	SSLasso ${\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ S-Z	0.047	0.063	0.076	0.063	0.090	0.099	0.053	0.076	0.121
Powerweak	dScore ${\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ S-Z	0.021	0.037	0.047	0.036	0.060	0.044	0.034	0.050	0.083
	nScore ${\lambda }_{\mathrm{s}\mathrm{u}\mathrm{p}}$ S-Z	0.071	0.089	0.136	0.081	0.121	0.104	0.114	0.113	0.123
	nScore ${\lambda }_{\text{1SE}}$ S-Z	0.039	0.060	0.050	0.076	0.074	0.066	0.070	0.067	0.104
	nScore ${\lambda }_{1\mathrm{S}\mathrm{E}}$ T2.2	0.056	0.070	0.073	0.093	0.087	0.080	0.107	0.096	0.111
	LDPE ${\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ S-Z	0.050	0.051	0.051	0.050	0.049	0.051	0.051	0.050	0.047
	SSLasso ${\lambda }_{\text{min}}$ S-Z	0.056	0.056	0.056	0.054	0.055	0.053	0.053	0.054	0.054
T1 Error	dScore ${\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ S-Z	0.033	0.036	0.034	0.031	0.031	0.035	0.036	0.035	0.033
	nScore ${\lambda }_{\text{sup}}$ S-Z	0.065	0.080	0.093	0.064	0.084	0.088	0.070	0.071	0.054
	nScore ${\lambda }_{\text{1SE}}$ S-Z	0.056	0.060	0.045	0.061	0.051	0.040	0.058	0.048	0.047
	nScore ${\lambda }_{1\text{SE}}$ T2.2	0.049	0.053	0.051	0.060	0.061	0.049	0.066	0.049	0.048