Prior-Shift-Invariant Threshold Transport in Budget-Constrained Supervisory Screening: Workload Identity, Prior–Drift Decomposition, and Distribution-Free Risk Control

Abstract

Examination capacity is fixed, so a supervisor screening firms must decide which files to open rather than which firms are deficient. A movement in the class prior rescales the posterior monotonically, leaving the ranking untouched and with it, the area under the receiver operating characteristic curve; that much is classical. This paper takes up the two quantities such a supervisor actually feels, neither of them a functional of the ranking. Hold the cut-off fixed: the detection rate does not move, the flagged share moves by the prevalence change times the Youden index—the adjusted-count identity of the quantification literature, read here as a workload forecast—and an exact algebraic split separates the deployment precision gap into a prior term, which recalibration reaches, and a residual, which it does not. Sorting the usual summaries by prior-invariance and by dependence on the operating region shows that both properties belong to quantities indexed by a fixed cut-off alone; the area under the curve carries only one, and precision and lift at a fixed budget only the other. The same indexing settles the decision problem: under a pure prior movement, carrying a cut-off across the deployment gap weakly dominates carrying a selection rate whichever way the prior moves, with an exact shortfall and a characterised equality case. That dominance is the paper’s one new analytical statement; the remaining apparatus is classical or a specialisation of existing methods, and a dedicated subsection of this paper says which is which, result by result. An exact binomial tolerance bound calibrates the budget—an order-statistic construction borrowed from Neyman–Pearson classification—and its certified level inflates by the distance between the positive-class score laws, a distance to which a prior movement contributes nothing. Two panels supply the evidence, and are the substance of the contribution: 122,335 United States firm-years under enforcement-based misstatement labels, and 172,566 bank-years over which the prior moves by a factor of 45 in both directions. Precision at the five per cent operating budget tracks the base rate there with an elasticity of 0.986, while chance-adjusted discrimination moves at −0.005, and a transported cut-off captures 71.4 and 93.1 per cent of the attainable audit surplus—the share of the value an oracle policy could realise at the same budget—against 2.8 and 33.2 for a standing fixed-share policy. Two negative findings follow: recalibration reaches only the smaller part of the deployment precision gap on the enforcement panel, and distribution-free certificates fall short of their certified coverage once deployed.