Algorithm 2.

The procedure that accepts original hypotheses while controlling type-II MCER

Input: matrix $I$, nominal FDR $\varphi$, nominal type-II MCER ${\alpha }_2$

Step 1: construct a $\left(1-{\tilde{\alpha }}_2\right)$-level one-sided $\mathrm{C}\mathrm{I}\left[0,{\tau }_u\right]$ for ${\tau }^{\mathrm{*}}$.

Calculate $(X_1,X_2,\dots ,X_m)$, $({\hat{p}}_1,{\hat{p}}_2,\dots ,{\hat{p}}_m)$, and then $\widehat{F}$.

For each bootstrap replicate $b=1,2,\dots ,B$,

Sample $n$ columns from $I$ with replacement to form $I^{\left(b\right)}$.

Calculate $(X_1^{\left(b\right)},X_2^{\left(b\right)},\dots ,X_m^{\left(b\right)})$, $(p_{-c,1}^{\left(b\right)},p_{-c,2}^{\left(b\right)},\dots ,p_{-c,m}^{\left(b\right)})$, and then $F_{-c}^{\left(b\right)}$.

Find $c_u^{\left(b\right)}$ to be the smallest $c$ that guarantees $\widehat{F}\prec F_{-c}^{\left(b\right)}$.

Find $c_u$, the smallest $c$ that satisfies (A3), to be the $\left(1-{\tilde{\alpha }}_2\right)$ quantile of $(c_u^{\left(1\right)},c_u^{\left(2\right)},\dots ,c_u^{\left(B\right)})$.

Find the upper limit ${\tau }_u$ to be the BH cutoff of p-values in (A2) indexed by $c_u$.

Step 2: Test hypotheses in (A1) given ${\tau }_u$ while controlling the FWER at $\left({\alpha }_2-{\tilde{\alpha }}_2\right)$.

Set $i=m$.

While $i\ge 1$,

If (A4) does not hold, accept all remaining hypotheses ${\tilde{H}}_{\left(i\right),0}^{\prime }\mathrm{s}$ and exit the loop.

Otherwise, reject ${\tilde{H}}_{\left(i\right),0}^{\prime }$, update $i=i-1$.

Whenever ${\tilde{H}}_{\left(i\right),0}^{\prime }$ is rejected, $H_{\left(i\right),0}$ is accepted.

Output: a decision for every hypothesis $H_{i,0}$ between “accepted” and “non-accepted”.