Fisher sharp null and predeclared test orderings

In the finite randomized treatment-assignment model Finite randomized treatment-assignment models, a schedule satisfies the Fisher sharp null when for each unit $i$ there is a value $y_i$ such that $Y_i(z)=y_i$ for every allowed assignment $z$. The full vector $y=(y_i{)}_{i\in \mathcal{I}}$ is then imputable under every assignment.

A predeclared ordering score is a real-valued statistic $T:\mathcal{Z}\times {\mathbb{R}}^{\mathcal{I}}\to \mathbb{R}$ (Statistics and estimators) fixed before the assignment and outcomes are inspected. Its upper-tail ordering includes ties: assignment $z^{\mathrm{\prime }}$ is at least as extreme as $z$ when $T(z^{\mathrm{\prime }},y)\ge T(z,y)$. For a signed score $S$ and fixed null center $s_0$, the corresponding two-sided ordering is explicitly $T(z,y)=\mathrm{\mid }S(z,y)-s_0\mathrm{\mid }$. The center is part of the predeclared ordering; the sharp null alone does not force $s_0=0$ for an arbitrary signed score. A lower-tail ordering is represented by using $-S$ as the score. A statistic in this setting is a measurable map in the sense of Statistics and estimators; the full power-set data sigma-algebra makes every such real-valued map measurable.