Bounding discretization error in worst-case randomization p-value decomposition

Establish a nontrivial upper bound—ideally one that vanishes as the maximum grid interval length tends to zero—for the discretization term arising in the Step 1 partition-based decomposition used to upper-bound the worst-case randomization p-value in the always-reporter testing procedure. Specifically, for complete randomization designs and any of the test statistics T_n^0, T_n^1, or T_n^2 defined for the always-reporter subpopulation, determine a useful bound on the probability that the randomization-distribution draw of the statistic falls within the grid interval containing the observed value, so that the gap between the computed maximal subproblem value and the true worst-case p-value can be controlled even when the randomization distribution is discrete with point masses.

Background

In the integer-programming approach for computing worst-case randomization p-values, the authors partition the statistic’s range and define subproblems whose optimal values yield an upper bound on the worst-case p-value. They show that the worst-case p-value is bounded above by the maximum subproblem value plus an additional term corresponding to the probability that the randomization-drawn statistic lies within the grid interval containing the observed value.

For continuous distributions, standard dominated convergence arguments would imply that this additional term vanishes as the grid is refined. However, in this setting the randomization distribution is discrete with point masses, so continuity-based arguments do not apply. The authors therefore note the lack of a simple refinement that provides a useful bound on this term, leaving a gap between the computable upper bound and the true worst-case p-value.

References

Ideally, we would like to show that, as the maximum interval length tends to zero, the additional term on the right-hand side of eqn:bound38 also tends to zero. If the underlying distribution were continuous, this would follow from an application of the dominated convergence theorem. In our setting, however, the randomization distributions are discrete with point masses, so such arguments are not directly applicable. We are not aware of a simple refinement that yields a useful bound for this term.

eqn:bound38:

$p^{\textrm{worst}} \leq \max_{i\in [I]}v_i \leq p^{\textrm{worst}} +E_{D\sim \textrm{CR}(n,n_1)}\left[\mathds{1}\{T_n(Y,D,A^{i^*})\in [t_{i^*-1},t_i^*)\}\right]. $

— Randomization Inference For the Always-Reporter Average Treatment Effect  (2603.24970 - Chang et al., 26 Mar 2026) in Section 6.2.1 (Step 1: Decomposing into subproblems), following Lemma 1 (Lemma \ref{lemma:v}) and inequality (38)