Statistical process control via $p$-values

TL;DR

This work proposes a model-free statistical process control framework that monitors processes using sequences of valid $p$-values under the in-control hypothesis via the super-uniformity property. It derives universal lower bounds for the average run length (ARL) and its multialarm generalisation (the $k$-ARL) that hold without dependence assumptions, and sharpens them to $1/\alpha$ and $k/\alpha$ under conditional super-uniformity. The authors introduce EWMA-like charts built from merging functions of $p$-values, providing smoothing without ad hoc control limits, and extend the approach to KS-based nonparametric monitoring and to multivariate directional localisation via closed testing with strong FWER control. The framework offers distribution-free calibration and modular extensions for smoothing and localisation, demonstrated through elementary, KS-based, and multivariate examples, with an accompanying open-source codebase. This p-value charting paradigm enables robust, interpretable SPC across VSI/VSS settings and heavy-tailed data, reducing reliance on parametric models and simulation-based calibration.

Abstract

We study statistical process control (SPC) through charting of $p$-values. When in control (IC), any valid sequence $(P_{t})_{t}$ is super-uniform, a requirement that can hold in nonparametric and two-phase designs without parametric modelling of the monitored process. Within this framework, we analyse the Shewhart rule that signals when $P_{t}\leα$. Under super-uniformity alone, and with no assumptions on temporal dependence, we derive universal IC lower bounds for the average run length (ARL) and for the expected time to the $k$th false alarm ($k$-ARL). When conditional super-uniformity holds, these bounds sharpen to the familiar $α^{-1}$ and $kα^{-1}$ rates, giving simple, distribution-free calibration for $p$-value charts. Beyond thresholding, we use merging functions for dependent $p$-values to build EWMA-like schemes that output, at each time $t$, a valid $p$-value for the hypothesis that the process has remained IC up to $t$, enabling smoothing without ad hoc control limits. We also study uniform EWMA processes, giving explicit distribution formulas and left-tail guarantees. Finally, we propose a modular approach to directional and coordinate localisation in multivariate SPC via closed testing, controlling the family-wise error rate at the time of alarm. Numerical examples illustrate the utility and variety of our approach.

Figures (2)

• Figure 1: Plots of PDFs of the random variables $\tilde{U}_{\lambda,t}$ with initialisation $u_{0}=1/2$, for $\lambda\in\left\{ 0.3,0.5,0.7\right\}$ and $t\in\left\{ 2,3,4\right\}$ along with histograms of 10000 replicates of the corresponding variable.
• Figure 2: Plots of CDFs of the random variables $\tilde{U}_{\lambda,t}$ with initialisation $u_{0}=1/2$, i.e., $F\left(\alpha\right)=\mathrm{P}_{0}\left(\tilde{U}_{\lambda,t}\le\alpha\right)$, for $\lambda\in\left\{ 0.3,0.5,0.7\right\}$ and $t\in\left\{ 2,3,4\right\}$ (solid line) along with the CDF of $U\sim\mathrm{Unif}\left[0,1\right]$ (dashed line) for $\alpha\le1/2$.

Theorems & Definitions (18)

• Proposition 1
• proof
• Proposition 2
• Proposition 3
• Proposition 4
• Lemma 5
• Proposition 6
• Proposition 7
• Remark 8
• Proposition 9