Phasing out Monte Carlo: exact acceptance probabilities for mean-spread release rules
Abstract
A large family of regulated release decisions accepts a production lot if and only if the pair (sample mean, sample standard deviation) of n units falls in a fixed plane region. Content uniformity (USP <905>), capability release, variables sampling and percent-within-limits highway acceptance all take this form. The acceptance probability under a hypothesized unit-level law is not, so far as we are aware, evaluated exactly in current practice: for a general parent the joint finite-n law of (X̅,s) is a constrained n-fold integral (Craig 1932 at n=3,4; Springer 1953 at general n) which admits no elementary reduction. Yet that probability is fixed by the law of the additive pair (T1,T2)=(∑i=1n Xi,∑i=1n Xi2), whose characteristic function is the n-th power of a one-observation transform. One two-dimensional Fourier inversion therefore delivers it for an arbitrary smooth parent - exactly, as an identity, and numerically on a grid whose dimension stays two whatever n is - a deterministic alternative to both the simulation and the Normality assumption. The identity is exact; what we evaluate is a finite-grid inversion of it, whose error we report per example rather than bound. At n=3 the Uniform parent yields the joint law of (X̅,s), the acceptance probability and the capability law in closed form; we derive these, use them as the anchor of a validation ladder, and then work three rules end to end under non-normal parents at their prescribed sample sizes (n=10-$30$). Against the exact answer the Normal-theory incumbent errs by enough to change the risk a plan is designed against, and errs in both directions along the operating curve, so no single constant corrects it. In general only the full distribution suffices.
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BibTeX
@misc{osan2026phasing,
title = {{Phasing out Monte Carlo: exact acceptance probabilities for mean-spread release rules}},
author = {Remus Osan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21577},
eprint = {2609.21577},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21577},
}
RIS
TY - UNPB
TI - Phasing out Monte Carlo: exact acceptance probabilities for mean-spread release rules
AU - Remus Osan
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21577
UR - https://arxiv.org/abs/2609.21577
AB - A large family of regulated release decisions accepts a production lot if and only if the pair (sample mean, sample standard deviation) of n units falls in a fixed plane region. Content uniformity (USP <905>), capability release, variables sampling and percent-within-limits highway acceptance all take this form. The acceptance probability under a hypothesized unit-level law is not, so far as we are aware, evaluated exactly in current practice: for a general parent the joint finite-n law of (X̅,s) is a constrained n-fold integral (Craig 1932 at n=3,4; Springer 1953 at general n) which admits no elementary reduction. Yet that probability is fixed by the law of the additive pair (T_1,T_2)=(∑_i=1^n X_i,∑_i=1^n X_i^2), whose characteristic function is the n-th power of a one-observation transform. One two-dimensional Fourier inversion therefore delivers it for an arbitrary smooth parent - exactly, as an identity, and numerically on a grid whose dimension stays two whatever n is - a deterministic alternative to both the simulation and the Normality assumption. The identity is exact; what we evaluate is a finite-grid inversion of it, whose error we report per example rather than bound. At n=3 the Uniform parent yields the joint law of (X̅,s), the acceptance probability and the capability law in closed form; we derive these, use them as the anchor of a validation ladder, and then work three rules end to end under non-normal parents at their prescribed sample sizes (n=10-$30$). Against the exact answer the Normal-theory incumbent errs by enough to change the risk a plan is designed against, and errs in both directions along the operating curve, so no single constant corrects it. In general only the full distribution suffices.
ER -