Enhancing the Detection Power of CUSUM Charts for Changes in the Mean of the Long-Memory ARFIMAX Model with Exponential White Noise
Received: 5 July 2025 | Revised: 27 July 2025, 22 August 2025, and 24 August 2025 | Accepted: 30 August 2025 | Online: 8 December 2025
Corresponding author: Wilasinee Peerajit
Abstract
The Cumulative Sum (CUSUM) control chart is highly effective in detecting small to moderate shifts in process means, making it a key tool for quality control. Its performance is commonly evaluated using the Average Run Length (ARL), defined as the expected number of samples before signaling an out-of-control condition. Traditional ARL estimation techniques, such as Monte Carlo simulations, Markov chains, and numerical Integral Equation (IE) methods, are computationally demanding and yield only approximate results. This paper presents a new explicit ARL computation method derived from the IE framework, with its validity established through Banach’s fixed point theorem. The method is applied to the CUSUM chart for monitoring a long-memory ARFIMAX process with exponential white noise. Benchmark comparisons with accurate but costly numerical IE methods (Midpoint, Trapezoidal, and Simpson’s Rules) indicate that the proposed method achieves near-perfect accuracy in a fraction of the time. Furthermore, a shift size of 0.75 provided the fastest detection, and a case study confirmed its effectiveness for real-time process monitoring.
Keywords:
Cumulative Sum (CUSUM) control chart, Average Run Length (ARL), long-memory ARFIMAX model, Banach’s Fixed-Point Theorem, numerical integral equation method, explicit methodDownloads
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