= Solution
A <predictive discrepancy statistic> is a specified measurable function $T$ of a possible dataset, chosen to detect a feature relevant to the <null hypothesis>. Under the fully specified null density $p_0$, compare the observed $T(\mathbf x)$ with its reference distribution, derived analytically or from replicated datasets $\mathbf X^{\mathrm{rep}}\sim p_0$.
For discrepancies whose large values indicate disagreement, use
$$
\boxed{p_{\mathrm{check}}=
\mathbb P_0\{T(\mathbf X^{\mathrm{rep}})\ge T(\mathbf x)\}.}
$$
A lower-tail or two-sided discrepancy requires the corresponding comparison. The checking function is a statistic, not a density; no unknown parameter is fitted when the null is fully specified.
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