Algorithm 2.

Adjusted simultaneous inference with FDR control and power enhancement.

Step 1: Initialization: Step 1.1: Compute the test statistics and auxiliary statistics {(Ti,, Ai); i = 1,…, q}. Step 1.2: Compute thep-values: pi = 2{1 − Φ(|Ti|)}, i,…,.q. Step 1.3: Input the pre-specified constants K, C1, C2 and N. Step 1.4: Compute the grid set: $$\mathcal{J}=\left\{\left(C_{1}N-1\right)\sqrt{\text{log\hspace{0.17em}}q}/N,C_1\sqrt{\text{log }q},\dots ,\left(C_{2}N-1\right)\sqrt{\text{log }q}/N,C_2\sqrt{\text{log }q}\right\}.$$ Step 2: For each ${\mathcal{J}}_K=\left\{{\lambda }_1,\dots ,{\lambda }_{K-1}\right\}$ in $\mathcal{J}$ and λo = −∞, λK = ∞: Step 2.1: Construct ${\mathcal{G}}_k=\left\{i:1\le i\le q,{\lambda }_{k-1}<A_i\le {\lambda }_k\right\},1\le k\le K$. Step 2.2: For each ${\mathcal{G}}_k$, compute the cardinality, $q_k=|{\mathcal{G}}_k|$. Step 2:3: For each ${\mathcal{G}}_k$, estimate the proportion, ${\widehat{\pi }}_k$, of alternatives in ${\mathcal{G}}_k$. Step 2.4: Compute the adjusting weights Wi, i = 1,…, q, according to (5). Step 2.5: Adjust the p-values: $p_i^w=\text{min}\left\{p_i/w_i,1\right\},i=1,\dots ,q$. Step 2.6: Apply the BH procedure, and record the total number of rejections. Step 3: Obtain the adjusted rejection region: Step 3.1: Choose ${\mathcal{J}}_K$ that yields the largest number of rejections. Step 3.2: Compute the corresponding adjusted p-values: $p_i^w,1\le i\le q$. Step 3.3: Reorder all the adjusted p- values: $p_{(1)}^w\le \dots \le p_{(q)}^w$. Step 3.4: Output the rejection region $\{i:i<\widehat{\tau }\}$, where $\widehat{\tau }=\text{max}\left\{i:p_{(i)}^w\le \alpha i/q\right\}$.