False discovery proportion, FDR, and FWER

Let $m\ge 1$ be a finite number of hypotheses and write $[m]=\{1,\dots ,m\}$. Fix a probability law $P$ and a set $I_0\subseteq [m]$ of indices whose null hypotheses are true under $P$. A measurable rejection rule is a measurable map $\mathcal{R}:\mathrm{\Omega }\to 2^{[m]}$, where the finite set $2^{[m]}$ has its discrete sigma-algebra. Set

$R=\mathrm{\mid }\mathcal{R}\mathrm{\mid },V=\mathrm{\mid }\mathcal{R}\cap I_0\mathrm{\mid }\mathrm{.}$

The false discovery proportion is

$\mathrm{FDP}:=\frac{V}{R\vee 1},R\vee 1:=\max (R,1),$

and the false discovery rate is

$\mathrm{FDR}:={\mathbb{E}}_P\text{\hspace{0.17em}⁣}\left[\frac{V}{R\vee 1}\right]\mathrm{.}$

The family-wise error rate is

$\mathrm{FWER}:=P(V\ge 1)\mathrm{.}$

Since $V\le R$ whenever $R>0$ and $V=0$ when $R=0$, the FDP is a measurable random variable in $[0,1]$ and its expectation is finite. These definitions use the fixed true-null set for the specified law; they do not assert that FDR and FWER are equal except in the global-null case.