Finite-sample optimal FDR-FNR frontier under the two-group model

Fix a finite integer $n \ge 1$. For each $i \in \{1,\dots,n\}$, let $\theta_i \in \{0,1\}$ be the latent hypothesis state, where $\theta_i=0$ means null and $\theta_i=1$ means non-null. Assume $(\theta_i)_{i=1}^n$ are i.i.d. with $\mathbb P(\theta_i=1)=\pi_1 \in (0,1)$ and $\pi_0=1-\pi_1$. Conditional on $\theta_i$, the test statistic $X_i$ takes values in a measurable space $(\mathcal X,\mathcal A)$ and has distribution

where $P_0,P_1$ are known and dominated by a common measure (densities $f_0,f_1$ may be used). Assume conditional independence across $i$: given $(\theta_1,\dots,\theta_n)$, the variables $X_1,\dots,X_n$ are independent.

This setup follows Nie & Wu (2023).

A (possibly compound, randomized) multiple-testing rule is any measurable map $\delta_n$ that, from $(X_1,\dots,X_n)$ and an auxiliary random seed independent of the data, outputs decisions $D_i \in \{0,1\}$ ($D_i=1$ means reject $H_i$). Define

The false discovery rate and false non-discovery rate are

where expectations are under the full two-group model (including rule randomization).

For $\alpha\in(0,1)$, define the finite-sample constrained objective