TABLE 1.

Coverage proportions (Cov) and average lengths (Len) of 95% naïve confidence intervals with tuning parameters ${\lambda }_{sup}$ and ${\lambda }_{\mathit{1}SE}$, and 95% exact post-selection confidence intervals under the scale-free graph setting with partial correlation $\rho \in \{0.2,0.6\}$, sample size $n\in \{100,300,500\}$, dimension $p=100$ and signal-to-noise ratio $\mathrm{S}\mathrm{N}\mathrm{R}\in \{0.1,0.3,0.5\}$. The error variance is estimated either through the scaled lasso (Sun and Zhang, 2012) (SL) or Theorem 2.2 (NL).

	$\rho$ $n$ SNR	0.1	100 0.3	0.5	0.1	0.2 300 0.3	0.5	0.1	500 0.3	0.5
	exact ${\lambda }_{\text{sup}}$ SL	0.905	0.942	0.936	0.949	0.953	0.953	0.946	0.949	0.949
Cov	naïve ${\lambda }_{\text{sup}}$ SL	0.683	0.961	0.944	0.951	0.952	0.951	0.971	0.948	0.947
	naïve ${\lambda }_{1\mathrm{S}\mathrm{E}}$ SL	0.596	0.876	0.887	0.905	0.917	0.910	0.944	0.935	0.932
	naïve ${\lambda }_{1\mathrm{S}\mathrm{E}}$ NL	0.587	0.870	0.883	0.905	0.919	0.911	0.945	0.939	0.933
	exact ${\lambda }_{\text{sup}}$ SL	4.260	1.634	0.823	1.907	0.437	0.330	0.814	0.327	0.255
Len	naïve ${\lambda }_{\text{sup}}$ SL	1.122	0.693	0.552	0.717	0.421	0.326	0.562	0.325	0.253
	naïve ${\lambda }_{\text{1SE}}$ SL	1.199	0.705	0.555	0.718	0.420	0.326	0.560	0.326	0.253
	naïve ${\lambda }_{1\mathrm{S}\mathrm{E}}$ NL	1.201	0.711	0.559	0.723	0.424	0.329	0.563	0.328	0.255
$\rho$ $n$ SNR		0.1	100 0.3	0.5	0.1	0.6 300 0.3	0.5	0.1	500 0.3	0.5
	exact ${\lambda }_{\text{sup}}$ SL	0.956	0.938	0.945	0.922	0.944	0.945	0.948	0.945	0.947
Cov	naïve ${\lambda }_{\text{sup}}$ SL	0.625	0.943	0.953	0.942	0.942	0.942	0.964	0.944	0.944
	naïve ${\lambda }_{1\mathrm{S}\mathrm{E}}$ SL	0.643	0.873	0.876	0.918	0.934	0.929	0.954	0.941	0.935
	naïve ${\lambda }_{1\mathrm{S}\mathrm{E}}$ NL	0.639	0.870	0.873	0.919	0.934	0.929	0.957	0.944	0.938
	exact ${\lambda }_{\text{sup}}$ SL	6.433	1.616	0.770	1.892	0.429	0.328	0.787	0.326	0.253
Len	naïve ${\lambda }_{\text{sup}}$ SL	1.108	0.692	0.553	0.713	0.420	0.326	0.560	0.325	0.252
	naïve ${\lambda }_{1\mathrm{S}\mathrm{E}}$ SL	1.200	0.705	0.554	0.720	0.420	0.326	0.561	0.326	0.253
	naïve ${\lambda }_{1\mathrm{S}\mathrm{E}}$ NL	1.203	0.709	0.557	0.725	0.423	0.329	0.564	0.327	0.254