Department of Applied Statistics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok, Thailand.
PLoS One. 2022 Feb 25;17(2):e0264283. doi: 10.1371/journal.pone.0264283. eCollection 2022.
The CUSUM control chart is suitable for detecting small to moderate parameter shifts for processes involving autocorrelated data. The average run length (ARL) can be used to assess the ability of a CUSUM control chart to detect changes in a long-memory seasonal autoregressive fractionally integrated moving average with exogenous variable (SARFIMAX) process with underlying exponential white noise. Herein, new ARLs via an analytical integral equation (IE) solution as an analytical IE and a numerical IE method to test a CUSUM control chart's ability to detect a wide range of shifts in the mean of a SARFIMAX(P, D, Q, r)s process with underlying exponential white noise are presented. The analytical IE formulas were derived by using the Fredholm integral equation of the second type while the numerical IE method for the approximate ARL is based on quadrature rules. After applying Banach's fixed-point theorem to guarantee its existence and uniqueness, the precision of the proposed analytical IE ARL was the same as the numerical IE method. The sensitivity and accuracy of the ARLs based on both methods were assessed on a CUSUM control chart running a SARFIMAX(P, D, Q, r)s process with underlying exponential white noise. The results of an extensive numerical study comprising the examination of a wide variety of out-of-control situations and computational schemes reveal that none of the methods outperformed the IE. Specifically, the computational scheme is easier and can be completed in one step. Hence, it is recommended for use in this situation. An illustrative example based on real data is also provided, the results of which were found to be in accordance with the research results.
CUSUM 控制图适用于检测涉及自相关数据的过程中的小到中等参数偏移。平均运行长度 (ARL) 可用于评估 CUSUM 控制图检测具有潜在指数白噪声的长记忆季节性自回归分数阶积分移动平均外生变量 (SARFIMAX) 过程中参数变化的能力。在此，通过分析积分方程 (IE) 解作为分析 IE 和数值 IE 方法，提出了新的 ARL，以测试 CUSUM 控制图检测 SARFIMAX(P, D, Q, r)s 过程均值广泛变化的能力，该过程具有潜在指数白噪声。分析 IE 公式是通过使用第二类弗雷德霍姆积分方程推导出来的，而数值 IE 方法用于近似 ARL 是基于求积规则。应用巴拿赫不动点定理来保证其存在性和唯一性之后，所提出的分析 IE ARL 的精度与数值 IE 方法相同。在具有潜在指数白噪声的 SARFIMAX(P, D, Q, r)s 过程上运行 CUSUM 控制图，评估了基于这两种方法的 ARL 的灵敏度和准确性。广泛的数值研究结果包括对各种失控情况和计算方案的检查，结果表明，没有一种方法优于 IE。具体来说，计算方案更简单，并且可以一步完成。因此，建议在这种情况下使用它。还提供了一个基于实际数据的说明性示例，结果与研究结果一致。
