Table 2.

We report the rejection frequencies of each method for the (uniform) confidence bands for $\mathcal{U}\times \tilde{p}$. The proposed estimator computes the critical value based on the multiplier bootstrap procedure. For the naive post-selection estimator, we report the results for two choices of critical values, one choice based on the multiplier bootstrap (MB) and another based on Bonferroni (BF) correction. The results are based on 500 replications

p = 2000, n = 500		Uniform over $\mathcal{U}\times \tilde{p}$
Design	Method	[1,2.5] × {1}	{1} × [10]	[1,2.5] × [10]	{1} × [1000]
1(i)	Proposed	0.054	0.036	0.048	0.040
	Naive (MB)	0.126	0.136	0.172	0.032
	Naive (BF)	0.014	0.124	0.026	0.032
1(ii)	Propose	0.270	0.036	0.032	0.142
	Naive (MB)	0.014	0.802	0.934	0.404
	Naive (BF)	0.000	0.802	0.718	0.376
Design	Method	[−0.5, 0.5] × {1}	{0.5} × [10]	[−0.5,0.5] × [10]	{0.5} × [1000]
2(i)	Proposed	0.364	0.038	0.052	0.062
	Naive (MB)	0.116	0.040	0.022	0.048
	Naive (BF)	0.018	0.038	0.000	0.046
2(ii)	Proposed	0.140	0.090	0.408	0.084
	Naive (MB)	0.002	0.946	0.996	0.362
	Naive (BF)	0.000	0.946	0.944	0.298