Valid p-values

Let $\mathsf{E}=((\mathcal{X},\mathcal{A}),\mathrm{\Theta },(P_{\theta }{)}_{\theta \in \mathrm{\Theta }})$ be a statistical experiment (Statistical experiments). A p-value statistic is a statistic $p:\mathcal{X}\to [0,1]$, where $[0,1]$ carries the trace of the Borel sigma-algebra, meaning $\{B\cap [0,1]:B\in \mathcal{B}(\mathbb{R})\}$ (Statistics and estimators, A measurable function between measurable spaces).

For a nonempty null hypothesis ${\mathrm{\Theta }}_0\subseteq \mathrm{\Theta }$, the statistic is a valid p-value for ${\mathrm{\Theta }}_0$ if it is superuniform under every null law: $P_{\theta }(\{x:p(x)\le u\})\le u\text{for every }\theta \in {\mathrm{\Theta }}_0\text{ and }u\in [0,1]\mathrm{.}$ The variable $p(X)$ is the observed p-value when the data are $X$. The validity condition is distributional; measurability alone does not imply it.

A discrete valid p-value may be conservative, so equality need not hold in the superuniform bound. Validity is required under every parameter in a composite null.