[Paper Review] Statistical process control via $p$-values

Hien Duy Nguyen, Dan Wang|arXiv (Cornell University)|Jan 24, 2026

Advanced Statistical Process Monitoring0 citations

TL;DR

The paper develops p-value based SPC charts, derives universal and conditional ARL bounds, builds EWMA-like p-value schemes, and enables multivariate localisation via closed testing.

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.

Motivation & Objective

• Motivate SPC monitoring via p-values and nonparametric design.
• Derive average run length (ARL) lower bounds under super-uniformity.
• Derive k-alarm run length (k-ARL) bounds and calibration insights.
• Develop EWMA-like schemes that operate directly on p-values.
• Propose a modular closed-testing approach for directional localisation in multivariate SPC.

Proposed method

• Define IC/OC framework using super-uniform p-values under H0.
• Prove ARL lower bounds: ARL ≥ (1/(2α)) + 1/2 without dependence assumptions.
• Strengthen ARL to ARL ≥ 1/α under conditional super-uniformity.
• Establish k-ARL bounds and discuss their sharpness.
• Construct EWMA-like p-value charts using p-value averaging with r and λ parameters.
• Provide a closed-testing based method for directional and coordinate localisation in multivariate SPC.

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.

Experimental results

Research questions

• RQ1 How can SPC be effectively monitored using p-values without strict dependence assumptions?
• RQ2 What are the universal and conditional ARL bounds for p-value charts under H0?
• RQ3 How can EWMA-like schemes be built directly from p-values while preserving validity?
• RQ4 How can multivariate SPC achieve directional and coordinate localisation with error control?

Key findings

• ARL for a p-value chart with P_t ≤ α is bounded below by (1/2α) + 1/2 under super-uniformity.
• Under conditional super-uniformity, ARL improves to at least 1/α.
• For k alarms, the k-ARL bound is at least (k/α) under the conditional assumption and is tight.
• EWMA-like schemes for p-values are feasible via p-value averaging with validity guarantees.
• A modular closed-testing framework enables directional and coordinate localisation in multivariate SPC with FWER control at alarms.
• The framework yields distribution-free calibration of p-value charts and avoids ad hoc control limits.

Figure 2: Plots of CDFs of the random variables $\tilde{U}_{\lambda,t}$ with initialisation $u_{0}=1/2$ , i.e., $F\left(\alpha\right)=\mathrm{P}_{0}\left(\tilde{U}_{\lambda,t}\leq\alpha\right)$ , for $\lambda\in\left\{0.3,0.5,0.7\right\}$ and $t\in\left\{2,3,4\right\}$ (solid line) along with the CD