 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, useA 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.