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 $(\bar X,s)$ 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 $(T_1,T_2)=(\sum_{i=1}^{n} X_i,\sum_{i=1}^{n} X_i^2)$, 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 $(\bar X,s)$, 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.