Algorithm 2.

Selection of $\tau$ for multiple testing.

1.	For any given $\tau$ and each $j\in 𝓗$, calculate ${\breve{\zeta }}_{j,\text{null}}(\tau )=\sum _{m=1}^M{n}_m{\left\{{\breve{\beta }}_{j,\text{null}}^{(\text{m})}(\tau )/{\widehat{\sigma }}_j^{(\text{m})}\right\}}^2$ with
	${\breve{\beta }}_{j,\text{null}}^{(\text{m})}(\tau )=K^{-1}\sum _{k=1}^K{(-1)}^{k>K/2}\left\{{\tilde{\beta }}_{j,[-k]}^{(\text{m})}+{\widehat{\mathit{u}}}_{j,[k]}^{(\text{m})\text{T}}(\tau )\left({\tilde{\mathit{\xi }}}_{[k]}^{(\text{m})}-{\widetilde{\mathrm{ℍ}}}_{[k]}^{(\text{m})}{\tilde{\mathit{\beta }}}_{[-k]}^{(\text{m})}\right)\right\},$
	where ${\widehat{\mathit{u}}}_{j,[k]}^{(•)}(\tau )$ is the debiasing projection direction obtained at tuning value $\tau$.
2.	Define ${\widehat{R}}_{0,\text{null}}(t\mid \tau )=\sum _{j\in 𝓗}I\left[{\overline{\mathbb{F}}}_{{\chi }_M^2}\left\{{\breve{\zeta }}_{j,\text{null}}(\tau )\right\}\le 2\overline{\text{Φ}}(t)\right]$ and a modified measure
	$\widehat{d}(\tau )={\int }_0^1{\left[{\widehat{R}}_{0,\text{null}}\left\{{\overline{\text{Φ}}}^{-1}(x)\mid \tau \right\}/(2qx)-1\right]}^{2}d\widehat{\omega }(x),$
	where $\widehat{\omega }(x)=H^{-1}\sum _{h=1}^{H}I\left(\overline{\text{Φ}}\left\{{(2\log q)}^{1/2}\right\}h/H\le x\right)$ and $H>0$ is some specified constant.