Derive the exact finite-sample coverage law at general dependence

Derive the exact finite-sample distribution of realised coverage for clustered calibration scores at arbitrary within-cluster dependence, beyond the independent and perfectly tied endpoints.

Background

Theorem 1 gives only an asymptotic normal limit for realised coverage, while the moment-matched Beta distribution used elsewhere is explicitly identified as a working approximation rather than a theorem. Exact laws are available in the independent case and at the perfectly tied endpoint, but the interior dependence regime remains unresolved.

A general finite-sample law would explain the deviations from the Beta approximation, including ceiling effects at perfect dependence, and would provide a principled replacement for the empirically validated but unproved finite-sample approximation.

References

The exact finite-sample law at general ρ is open, and the ρ = 1 case shows why it is hard: the tied law is governed by ⌈k/m⌉ ceiling discreteness that no smooth function of ρ reproduces.

— The Exceedance Design Effect: Effective Sample Size for Thresholds under Clustering  (2608.21262 - Noonan, 21 Aug 2026) in Section 10, p. 34

What the data establish needs no λU : the exceedance correlation at p = 0.99 is 0.228 [0.172, 0.283]. Whether it plateaus there or continues toward zero beyond 0.99 is open, and settling it needs reference arms rebuilt at the measured ˜m and ρI .

— The Exceedance Design Effect: Effective Sample Size for Thresholds under Clustering  (2608.21262 - Noonan, 21 Aug 2026) in Section 6.2, p. 22

What remains open is the composition, and the two do not compose neutrally. They pull against each other, through the same cluster sizes: reweighting to the cluster-average marginal carries a Kish weighting effective size that never exceeds n, costing 19.9% of the calibration set on §6.1’s released profile and climbing with raggedness to a ratio of 0.26 at CV2 = 2.1 (reweighting_cost.py).

— The Exceedance Design Effect: Effective Sample Size for Thresholds under Clustering  (2608.21262 - Noonan, 21 Aug 2026) in Section 10, p. 35