{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T19:30:37Z","timestamp":1780601437536,"version":"3.54.1"},"reference-count":23,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,11]],"date-time":"2025-01-11T00:00:00Z","timestamp":1736553600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Thailand Science Research and Innovation Fund (NSRF) and King Mongkut\u2019s University of Technology North Bangkok","award":["KMUTNB-FF-68-B-08"],"award-info":[{"award-number":["KMUTNB-FF-68-B-08"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Control charts are frequently used instruments for process quality monitoring. Another name for the NEEWMA control chart is the new extended exponentially weighted moving average (new extended EWMA) control chart. The lower control limit (LCL) and upper control limit (UCL) are equally spaced from the center line, giving it a symmetrical design. Because of its symmetry, the NEEWMA chart is very good at identifying even the tiniest changes in operation by detecting deviations from the target in both upward and downward directions. This study derives an explicit formula for the average run length (ARL) of the NEEWMA control chart based on the autoregressive (AR) model with exponential white noise. The focus is on the zero-state performance of the NEEWMA control chart, which is derived using explicit formulas. Banach\u2019s fixed-point theorem was used to prove existence and uniqueness of this formula. The accuracy of this formula is validated by comparing it to the numerical integral equation (NIE) method using percentage accuracy (%Acc). The results show that the NEEWMA control chart is more efficient than the ARL evaluated by the NIE method, particularly regarding computation time. The performance of the NEEWMA control chart is compared with the EWMA and extended EWMA control charts by evaluating both the ARL and standard deviation run length (SDRL). The NEEWMA control chart outperforms the others in detection performance, followed by the extended EWMA and EWMA control charts. Further verification of its superior performance is provided through comparisons using the average extra quadratic loss (AEQL) and the performance comparison index (PCI), which confirm that it outperforms both the EWMA and extended EWMA control charts across various parameters and shift sizes. Finally, an illustrative example using real-life economic data demonstrates its efficiency.<\/jats:p>","DOI":"10.3390\/sym17010104","type":"journal-article","created":{"date-parts":[[2025,1,13]],"date-time":"2025-01-13T04:01:52Z","timestamp":1736740912000},"page":"104","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["The Efficiency of the New Extended EWMA Control Chart for Detecting Changes Under an Autoregressive Model and Its Application"],"prefix":"10.3390","volume":"17","author":[{"given":"Kotchaporn","family":"Karoon","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand"},{"name":"Research Center for Academic Excellence in Mathematics, Naresuan University, Phitsanulok 65000, Thailand"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5103-9867","authenticated-orcid":false,"given":"Yupaporn","family":"Areepong","sequence":"additional","affiliation":[{"name":"Department of Applied Statistics, Faculty of Applied Science, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1051\/ijmqe\/2018003","article-title":"Control chart applications in healthcare: A literature review","volume":"9","author":"Suman","year":"2018","journal-title":"Int. J. Metrol. Qual. Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"19595","DOI":"10.3934\/math.20231000","article-title":"Enhanced fault detection models with real-life applications","volume":"8","author":"Lone","year":"2023","journal-title":"AIMS Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"546","DOI":"10.1080\/01621459.1925.10502930","article-title":"The application of statistics as an aid in maintaining quality of a manufactured product","volume":"20","author":"Shewhart","year":"1925","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1093\/biomet\/41.1-2.100","article-title":"Continuous inspection schemes","volume":"41","author":"Page","year":"1954","journal-title":"Biometrika"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1080\/00401706.1959.10489860","article-title":"Control chart tests based on qeometric moving averages","volume":"1","author":"Roberts","year":"1959","journal-title":"Technometrics"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Naveed, M., Azam, M., Khan, N., and Aslam, M. (2018). Design of a control chart using extended EWMA statistic. Technologies, 6.","DOI":"10.3390\/technologies6040108"},{"key":"ref_7","first-page":"e34424","article-title":"New extended exponentially weighted moving average control chart for monitoring process mean","volume":"10","author":"Javed","year":"2024","journal-title":"Technometrics"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1080\/0233188031000078042","article-title":"Bayesian estimation of an AR(1) process with exponential white noise","volume":"37","author":"Ibazizen","year":"2003","journal-title":"Statistics"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1287\/mnsc.45.3.395","article-title":"Efficient design and sensitivity analysis of control charts using Monte Carlo simulation","volume":"45","author":"Fu","year":"1999","journal-title":"Manag. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1093\/biomet\/59.3.539","article-title":"An approach to the probability distribution of the CUSUM run length","volume":"59","author":"Brook","year":"1972","journal-title":"Biometrika"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1080\/03610919108812948","article-title":"A comparison of the Markov chain and the integral equation approaches for evaluating the run length distribution of quality control charts","volume":"20","author":"Champ","year":"1991","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Zhao, H., Tang, H., Pang, C., and Jiang, H. (2022). A Markov Chain Model for Approximating the Run Length Distributions of Poisson EWMA Charts under Linear Drifts. Mathematics, 10.","DOI":"10.3390\/math10244786"},{"key":"ref_13","first-page":"2143","article-title":"Average Run Length of Control Chart for ARX(1) Process with Exponential White Noise","volume":"12","author":"Paichit","year":"2016","journal-title":"Glob. J. Pure Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"88","DOI":"10.13189\/ms.2022.100107","article-title":"Explicit Formulas and Numerical Integral Equation of ARL for SARX(P,r)L Model Based on CUSUM Chart","volume":"10","author":"Phanyaem","year":"2022","journal-title":"Math. Stat."},{"key":"ref_15","first-page":"APST-27","article-title":"The average run length for continuous distribution process mean shift detection on a modified EWMA control chart","volume":"27","author":"Paichit","year":"2022","journal-title":"Asia-Pac. J. Sci. Technol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2219040","DOI":"10.1080\/09540091.2023.2219040","article-title":"Modification of ARL for detecting changes on the double EWMA chart in time series data with the autoregressive model","volume":"35","author":"Karoon","year":"2023","journal-title":"Connect. Sci."},{"key":"ref_17","first-page":"7088","article-title":"Performance measurement of a DMEWMA control chart on an AR(p) model with exponential white noise","volume":"17","author":"Phanthuna","year":"2024","journal-title":"Appl. Sci. Eng. Prog."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1601","DOI":"10.1134\/S1995080224601577","article-title":"Enhancing the performance of an adjusted MEWMA control chart running on trend and quadratic trend autoregressive models","volume":"45","author":"Karoon","year":"2024","journal-title":"Lobachevskii J. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"278","DOI":"10.13189\/ms.2020.080305","article-title":"Adomian decomposition method with modified Bernstein polynomials for solving nonlinear Fredholm and Volterra integral equations","volume":"8","author":"Almousa","year":"2020","journal-title":"Math. Stat."},{"key":"ref_20","unstructured":"Serrin, J. (2005). Gradient Estimates for Solutions of Nonlinear Elliptic and Parabolic Equations, Chapman and Hall\/CRC."},{"key":"ref_21","first-page":"627","article-title":"The Integral Equation Approach for Solving the Average Run Length of EWMA Procedure for Autocorrelated Process","volume":"19","author":"Phanyaem","year":"2021","journal-title":"Thail. Stat."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Fonseca, A., Ferreira, P.H., Nascimento, D.C., Fiaccone, R., Correa, C.U., Pi\u00f1a, A.G., and Louzada, F. (2021). Water particles monitoring in the Atacama desert: SPC approach based on proportional data. Axioms, 10.","DOI":"10.3390\/axioms10030154"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"532","DOI":"10.1002\/qre.2999","article-title":"The triple exponentially weighted moving average control chart for monitoring Poisson process","volume":"38","author":"Alevizakos","year":"2023","journal-title":"Qual. Reliab. Eng. Int."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/104\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:27:00Z","timestamp":1759919220000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/104"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,11]]},"references-count":23,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["sym17010104"],"URL":"https:\/\/doi.org\/10.3390\/sym17010104","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,11]]}}}