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題名 以符號檢定為基的多元比例管制圖
A multidimensional ratios control chart based on the sign test
作者 鄭智忠
Cheng, Chih-Chung
貢獻者 楊素芬<br>葉百堯
Yang, Su-Fen<br>Yeh, Arthur
鄭智忠
Cheng, Chih-Chung
關鍵詞 多維度管制圖
平均值比
符號檢定
平均連串長度
Average run length
Multidimensional ratios of means
Multivariate control chart
Sign test
日期 2022
上傳時間 2-九月-2022 14:46:00 (UTC+8)
摘要 在製程監控的領域中，管制圖是常見且有效的方法。
在許多產業中，如玻璃工業、化學工業以及食品業等行業，追蹤相關的製程變數平均值比或是成分間平均值的比例是個重要的課題。過去關於監控平均值比的管制圖研究，多集中於二元常態變數的平均值比，多維度平均值比之管制圖尚未被探討。
本研究提出一個可以監控各種分配多維度平均值比之管制圖。本文將符號檢定（sign test）的方法應用於多維度平均值比的監控，建立一個可以適用於各種分配的多維度平均值比管制圖，並以平均連串長度（ARL）做為衡量管制圖偵測失控製程的能力指標。我們發現此管制圖的偵測能力在相關係數大、變異係數小以及樣本數大時偵測能力較佳；在不同的失控分配中，數據分析顯示此管制圖在多元均勻分配下製程的偵測能力表現最佳，而在多元非中心t分配下製程的偵測能力表現最差。此管制圖在各種分配下的具有相似的管制界限，因此可以將管制界限取平均值，而建立一致的管制界限。此外，建立此管制圖的管制界限只需決定每組抽樣樣本數以及管制圖的加權常數即可。最後以牛奶成份資料說明所提出的多維度平均值比管制圖的應用。
In the manufacturing processes monitoring, control chart is a widely used and effective approach. Monitoring the multidimensional ratios of the means among correlated process variables or ingredients of a product is important in many industries, such as glass industry, chemistry industry and food industry. However, existing studies about the single ratio of means control chart all focus on bivariate normal variables. Thus, we are motivated to develop a new control chart for monitoring multidimensional ratios of means of correlated variables under any distributions.
In this article, we apply a sign test approach to monitor the multidimensional ratios of means, by constructing a distribution-free multidimensional ratios of means control chart. We use the average run length (ARL) to measure the detection performance of the proposed control chart. We find that the proposed control chart has good detection performance with large correlation coefficient, small coefficient of variation, and large sample size. Numerical analyses show that the proposed chart has the best detection performance under multivariate uniform distribution, and the worst detection performance under multivariate noncentral t distribution. The proposed chart has similar control limits under different distributions, hence, a unified control limit can be obtained by taking the average of the control limits. The unified control limit of the proposed control chart could be obtained given only the group sample size and the smoothing parameter for any distributed correlated quality variables. Data of the ingredients of milk are presented to demonstrate the application of the proposed multidimensional ratios of means control chart.
參考文獻 Abubakar, S. S., Khoo, M. B., Saha, S., & Teoh, W. L. (2020). Run sum control chart for monitoring the ratio of population means of a bivariate normal distribution. Communications in Statistics-Theory and Methods, 51(13), 4559-4588.
Åström, K. J., Hägglund, T., & Astrom, K. J. (2006). Advanced PID control (Vol. 461). Research Triangle Park: ISA-The Instrumentation, Systems, and Automation Society.
Atchinson, J. A. (2005, October). Concise Guide to Compositional Data Analysis. In In2do Compositional Data Analysis Workshop CoDaWork Oct (Vol. 5, pp. 17-21).
Alloway Jr, J. A., & Raghavachari, M. (1991). Control chart based on the Hodges-Lehmann estimator. Journal of Quality Technology, 23(4), 336-347.
Alt, F. B. (1985). Multivariate quality control. Encyclopedia of Statistical Science, 6, 110-122.
Amin, R. W., Reynolds Jr., M. R., & Saad, B. (1995). Nonparametric quality control charts based on the sign statistic. Communications in Statistics-Theory and Methods, 24(6), 1597-1623.
Bakir, S. T. (2004). A distribution-free Shewhart quality control chart based on signed-ranks. Quality Engineering, 16(4), 613-623.
Bell, R. C., Jones-Farmer, L. A., & Billor, N. (2014). A distribution-free multivariate phase I location control chart for subgrouped data from elliptical distributions. Technometrics, 56(4), 528-538.
Cedilnik, A., Košmelj, K., & Blejec, A. (2004). The distribution of the ratio of jointly normal variables. Advances in Methodology and Statistics, 1(1), 99-108.
Celano, G., & Castagliola, P. (2016). Design of a phase II control chart for monitoring the ratio of two normal variables. Quality and Reliability Engineering International, 32(1), 291-308.
Celano, G., Castagliola, P., Faraz, A., & Fichera, S. (2014). Statistical performance of a control chart for individual observations monitoring the ratio of two normal variables. Quality and Reliability Engineering International, 30(8), 1361-1377.
Chen, N., Zi, X., & Zou, C. (2016). A distribution-free multivariate control chart. Technometrics, 58(4), 448-459.
Crowder, S. V. (1987). A simple method for studying run–length distributions of exponentially weighted moving average charts. Technometrics, 29(4), 401-407.
Crowder, S. V. (1989). Design of exponentially weighted moving average schemes. Journal of Quality technology, 21(3), 155-162.
Farokhnia, M., & Niaki, S. T. A. (2020). Principal component analysis-based control charts using support vector machines for multivariate non-normal distributions. Communications in Statistics-Simulation and Computation, 49(7), 1815-1838.
Fox, J., & Weisberg, S. (2017). Examples of Effect Displays with Partial Residuals Using Contrived Regression Data.
https://cran.r-project.org/web/packages/effects/vignettes/partial-residuals.pdf
Graham, M. A., Chakraborti, S., & Human, S. W. (2011). A nonparametric exponentially weighted moving average signed-rank chart for monitoring location. Computational Statistics & Data Analysis, 55(8), 2490-2503.
Graham, M. A., Human, S. W., & Chakraborti, S. (2010). A Phase I nonparametric Shewhart-type control chart based on the median. Journal of Applied Statistics, 37(11), 1795-1813.
Haq, A., & Sohrab, K. (2021). Directionally sensitive MCUSUM mean charts. Quality and Reliability Engineering International, 37(5), 2169-2188.
Hettmansperger, T. P., & Randles, R. H. (2002). A practical affine equivariant multivariate median. Biometrika, 89(4), 851-860.
Hinkley, D. V. (1969). On the ratio of two correlated normal random variables. Biometrika, 56(3), 635-639.
Hotelling, H. (1947). Multivariate quality control. Techniques of Statistical Analysis.
Jones-Farmer, L. A., Jordan, V., & Champ, C. W. (2009). Distribution-free phase I control charts for subgroup location. Journal of Quality Technology, 41(3), 304-316.
Lee, R. Y., Holland, B. S., & Flueck, J. A. (1979). Distribution of a ratio of correlated gamma random variables. SIAM Journal on Applied Mathematics, 36(2), 304-320.
Li, S. Y., Tang, L. C., & Ng, S. H. (2010). Nonparametric CUSUM and EWMA control charts for detecting mean shifts. Journal of Quality Technology, 42(2), 209-226.
Liu, R. Y. (1995). Control charts for multivariate processes. Journal of the American Statistical Association, 90(432), 1380-1387.
Lowry, C. A., Woodall, W. H., Champ, C. W., & Rigdon, S. E. (1992). A multivariate exponentially weighted moving average control chart. Technometrics, 34(1), 46-53.
Lucas, J. M., & Saccucci, M. S. (1990). Exponentially weighted moving average control schemes: properties and enhancements. Technometrics, 32(1), 1-12.
Marsaglia, G. (1965). Ratios of normal variables and ratios of sums of uniform variables. Journal of the American Statistical Association, 60(309), 193-204.
Malela-Majika, J. C. (2021). New distribution-free memory-type control charts based on the Wilcoxon rank-sum statistic. Quality Technology & Quantitative Management, 18(2), 135-155.
Malela-Majika, J. C., Chakraborti, S., & Graham, M. A. (2016). Distribution-free Phase II Mann–Whitney control charts with runs-rules. The International Journal of Advanced Manufacturing Technology, 86(1), 723-735.
Nguyen, H. D., Tran, K. P., & Heuchenne, C. (2019). Monitoring the ratio of two normal variables using variable sampling interval exponentially weighted moving average control charts. Quality and Reliability Engineering International, 35(1), 439-460.
ÖKsoy, D., & Aroian, L. A. (1994). The quotient of two correlated normal variables with applications. Communications in Statistics-Simulation and Computation, 23(1), 223-241.
ÖKSOY, D., Boulos, E., & DAVID PYE, L. (1993). Statistical process control by the quotient of two correlated normal variables. Quality Engineering, 6(2), 179-194.
Park, C., & Reynolds Jr, M. R. (1987). Nonparametric procedures for monitoring a location parameter based on linear placement statistics. Sequential Analysis, 6(4), 303-323.
Pignatiello Jr, J. J., & Runger, G. C. (1990). Comparisons of multivariate CUSUM charts. Journal of Quality Technology, 22(3), 173-186.
Qiu, P. (2008). Distribution-free multivariate process control based on log-linear modeling. IIE Transactions, 40(7), 664-677.
Randles, R. H. (1989). A distribution-free multivariate sign test based on interdirections. Journal of the American Statistical Association, 84(408), 1045-1050.
Randles, R. H. (2000). A simpler, affine-invariant, multivariate, distribution-free sign test. Journal of the American Statistical Association, 95(452), 1263-1268.
Roberts, S. W. (2000). Control chart tests based on geometric moving averages. Technometrics, 1(3), 239-250.
Schott, J. R. (2002). Testing for elliptical symmetry in covariance-matrix-based analyses. Statistics & Probability Letters, 60(4), 395-404.
Shewhart, W. A. (1924). Some applications of statistical methods to the analysis of physical and engineering data. Bell System Technical Journal, 3(1), 43-87.
Spisak, A. W. (1990). A control chart for ratios. Journal of Quality Technology, 22(1), 34-37.
Tran, K. P., Castagliola, P., & Celano, G. (2016). Monitoring the ratio of two normal variables using run rules type control charts. International Journal of Production Research, 54(6), 1670-1688.
Tran, K. P., Castagliola, P., & Celano, G. (2016). Monitoring the ratio of two normal variables using EWMA type control charts. Quality and Reliability Engineering International, 32(5), 1853-1869.
Tran, K. P., Castagliola, P., & Celano, G. (2018). Monitoring the ratio of population means of a bivariate normal distribution using CUSUM type control charts. Statistical Papers, 59(1), 387-413.
Tran, K. P., & Knoth, S. (2018). Steady‐state ARL analysis of ARL‐unbiased EWMA‐RZ control chart monitoring the ratio of two normal variables. Quality and Reliability Engineering International, 34(3), 377-390.
Tyler, D. E. (1987). A distribution-free M-estimator of multivariate scatter. The Annals of Statistics, 234-251.
Wang, S., & Reynolds Jr, M. R. (2013). A GLR control chart for monitoring the mean vector of a multivariate normal process. Journal of Quality Technology, 45(1), 18-33.
Yang, S. F., Lin, J. S., & Cheng, S. W. (2011). A new nonparametric EWMA sign control chart. Expert Systems with Applications, 38(5), 6239-6243.
Yang, S. F., Lin, Y. C., & Yeh, A. B. (2021). A Phase II depth‐based variable dimension EWMA control chart for monitoring process mean. Quality and Reliability Engineering International, 37(6), 2384-2398.
Yen, C. L., Shiau, J. J. H., & Yeh, A. B. (2012). Effective control charts for monitoring multivariate process dispersion. Quality and Reliability Engineering International, 28(4), 409-426.
Yourstone, S. A., & Zimmer, W. J. (1992). Non‐normality and the design of control charts for averages. Decision sciences, 23(5), 1099-1113.
Zou, C., & Tsung, F. (2011). A multivariate sign EWMA control chart. Technometrics, 53(1), 84-97.
描述 碩士
國立政治大學
統計學系
109354012
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0109354012
資料類型 thesis
dc.contributor.advisor 楊素芬<br>葉百堯zh_TW
dc.contributor.advisor Yang, Su-Fen<br>Yeh, Arthuren_US
dc.contributor.author (作者) 鄭智忠zh_TW
dc.contributor.author (作者) Cheng, Chih-Chungen_US
dc.creator (作者) 鄭智忠zh_TW
dc.creator (作者) Cheng, Chih-Chungen_US
dc.date (日期) 2022en_US
dc.date.accessioned 2-九月-2022 14:46:00 (UTC+8)-
dc.date.available 2-九月-2022 14:46:00 (UTC+8)-
dc.date.issued (上傳時間) 2-九月-2022 14:46:00 (UTC+8)-
dc.identifier (其他 識別碼) G0109354012en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/141548-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 統計學系zh_TW
dc.description (描述) 109354012zh_TW
dc.description.abstract (摘要) 在製程監控的領域中，管制圖是常見且有效的方法。
在許多產業中，如玻璃工業、化學工業以及食品業等行業，追蹤相關的製程變數平均值比或是成分間平均值的比例是個重要的課題。過去關於監控平均值比的管制圖研究，多集中於二元常態變數的平均值比，多維度平均值比之管制圖尚未被探討。
本研究提出一個可以監控各種分配多維度平均值比之管制圖。本文將符號檢定（sign test）的方法應用於多維度平均值比的監控，建立一個可以適用於各種分配的多維度平均值比管制圖，並以平均連串長度（ARL）做為衡量管制圖偵測失控製程的能力指標。我們發現此管制圖的偵測能力在相關係數大、變異係數小以及樣本數大時偵測能力較佳；在不同的失控分配中，數據分析顯示此管制圖在多元均勻分配下製程的偵測能力表現最佳，而在多元非中心t分配下製程的偵測能力表現最差。此管制圖在各種分配下的具有相似的管制界限，因此可以將管制界限取平均值，而建立一致的管制界限。此外，建立此管制圖的管制界限只需決定每組抽樣樣本數以及管制圖的加權常數即可。最後以牛奶成份資料說明所提出的多維度平均值比管制圖的應用。		zh_TW
dc.description.abstract (摘要) In the manufacturing processes monitoring, control chart is a widely used and effective approach. Monitoring the multidimensional ratios of the means among correlated process variables or ingredients of a product is important in many industries, such as glass industry, chemistry industry and food industry. However, existing studies about the single ratio of means control chart all focus on bivariate normal variables. Thus, we are motivated to develop a new control chart for monitoring multidimensional ratios of means of correlated variables under any distributions.
In this article, we apply a sign test approach to monitor the multidimensional ratios of means, by constructing a distribution-free multidimensional ratios of means control chart. We use the average run length (ARL) to measure the detection performance of the proposed control chart. We find that the proposed control chart has good detection performance with large correlation coefficient, small coefficient of variation, and large sample size. Numerical analyses show that the proposed chart has the best detection performance under multivariate uniform distribution, and the worst detection performance under multivariate noncentral t distribution. The proposed chart has similar control limits under different distributions, hence, a unified control limit can be obtained by taking the average of the control limits. The unified control limit of the proposed control chart could be obtained given only the group sample size and the smoothing parameter for any distributed correlated quality variables. Data of the ingredients of milk are presented to demonstrate the application of the proposed multidimensional ratios of means control chart.		en_US
dc.description.tableofcontents 1. Introduction 1
2. Multivariate Sign Statistic-Based Multidimensional Ratios of Means Control Chart 6
2.1 Multivariate sign statistic 6
2.2 Multivariate sign EWMA control chart 8
2.3 Design of multidimensional ratios of means EWMA control chart 10
2.4 The procedure to determine the control limit and average run length of the EWMAMR control chart 11
3. UCL and Detection Performance of the Proposed Control Chart 13
3.1 The multidimensional ratios of correlated random variables is an elliptical symmetry distribution 15
3.2 Detection performances and corresponding control limits under different distributions 18
4. Detection Performance of the EWMAMR Control Chart with a Unified UCL 40
5. A Real Example 63
6. Conclusions 72
References 73		zh_TW
dc.format.extent 1560949 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0109354012en_US
dc.subject (關鍵詞) 多維度管制圖zh_TW
dc.subject (關鍵詞) 平均值比zh_TW
dc.subject (關鍵詞) 符號檢定zh_TW
dc.subject (關鍵詞) 平均連串長度zh_TW
dc.subject (關鍵詞) Average run lengthen_US
dc.subject (關鍵詞) Multidimensional ratios of meansen_US
dc.subject (關鍵詞) Multivariate control charten_US
dc.subject (關鍵詞) Sign testen_US
dc.title (題名) 以符號檢定為基的多元比例管制圖zh_TW
dc.title (題名) A multidimensional ratios control chart based on the sign testen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) Abubakar, S. S., Khoo, M. B., Saha, S., & Teoh, W. L. (2020). Run sum control chart for monitoring the ratio of population means of a bivariate normal distribution. Communications in Statistics-Theory and Methods, 51(13), 4559-4588.
Åström, K. J., Hägglund, T., & Astrom, K. J. (2006). Advanced PID control (Vol. 461). Research Triangle Park: ISA-The Instrumentation, Systems, and Automation Society.
Atchinson, J. A. (2005, October). Concise Guide to Compositional Data Analysis. In In2do Compositional Data Analysis Workshop CoDaWork Oct (Vol. 5, pp. 17-21).
Alloway Jr, J. A., & Raghavachari, M. (1991). Control chart based on the Hodges-Lehmann estimator. Journal of Quality Technology, 23(4), 336-347.
Alt, F. B. (1985). Multivariate quality control. Encyclopedia of Statistical Science, 6, 110-122.
Amin, R. W., Reynolds Jr., M. R., & Saad, B. (1995). Nonparametric quality control charts based on the sign statistic. Communications in Statistics-Theory and Methods, 24(6), 1597-1623.
Bakir, S. T. (2004). A distribution-free Shewhart quality control chart based on signed-ranks. Quality Engineering, 16(4), 613-623.
Bell, R. C., Jones-Farmer, L. A., & Billor, N. (2014). A distribution-free multivariate phase I location control chart for subgrouped data from elliptical distributions. Technometrics, 56(4), 528-538.
Cedilnik, A., Košmelj, K., & Blejec, A. (2004). The distribution of the ratio of jointly normal variables. Advances in Methodology and Statistics, 1(1), 99-108.
Celano, G., & Castagliola, P. (2016). Design of a phase II control chart for monitoring the ratio of two normal variables. Quality and Reliability Engineering International, 32(1), 291-308.
Celano, G., Castagliola, P., Faraz, A., & Fichera, S. (2014). Statistical performance of a control chart for individual observations monitoring the ratio of two normal variables. Quality and Reliability Engineering International, 30(8), 1361-1377.
Chen, N., Zi, X., & Zou, C. (2016). A distribution-free multivariate control chart. Technometrics, 58(4), 448-459.
Crowder, S. V. (1987). A simple method for studying run–length distributions of exponentially weighted moving average charts. Technometrics, 29(4), 401-407.
Crowder, S. V. (1989). Design of exponentially weighted moving average schemes. Journal of Quality technology, 21(3), 155-162.
Farokhnia, M., & Niaki, S. T. A. (2020). Principal component analysis-based control charts using support vector machines for multivariate non-normal distributions. Communications in Statistics-Simulation and Computation, 49(7), 1815-1838.
Fox, J., & Weisberg, S. (2017). Examples of Effect Displays with Partial Residuals Using Contrived Regression Data.
https://cran.r-project.org/web/packages/effects/vignettes/partial-residuals.pdf
Graham, M. A., Chakraborti, S., & Human, S. W. (2011). A nonparametric exponentially weighted moving average signed-rank chart for monitoring location. Computational Statistics & Data Analysis, 55(8), 2490-2503.
Graham, M. A., Human, S. W., & Chakraborti, S. (2010). A Phase I nonparametric Shewhart-type control chart based on the median. Journal of Applied Statistics, 37(11), 1795-1813.
Haq, A., & Sohrab, K. (2021). Directionally sensitive MCUSUM mean charts. Quality and Reliability Engineering International, 37(5), 2169-2188.
Hettmansperger, T. P., & Randles, R. H. (2002). A practical affine equivariant multivariate median. Biometrika, 89(4), 851-860.
Hinkley, D. V. (1969). On the ratio of two correlated normal random variables. Biometrika, 56(3), 635-639.
Hotelling, H. (1947). Multivariate quality control. Techniques of Statistical Analysis.
Jones-Farmer, L. A., Jordan, V., & Champ, C. W. (2009). Distribution-free phase I control charts for subgroup location. Journal of Quality Technology, 41(3), 304-316.
Lee, R. Y., Holland, B. S., & Flueck, J. A. (1979). Distribution of a ratio of correlated gamma random variables. SIAM Journal on Applied Mathematics, 36(2), 304-320.
Li, S. Y., Tang, L. C., & Ng, S. H. (2010). Nonparametric CUSUM and EWMA control charts for detecting mean shifts. Journal of Quality Technology, 42(2), 209-226.
Liu, R. Y. (1995). Control charts for multivariate processes. Journal of the American Statistical Association, 90(432), 1380-1387.
Lowry, C. A., Woodall, W. H., Champ, C. W., & Rigdon, S. E. (1992). A multivariate exponentially weighted moving average control chart. Technometrics, 34(1), 46-53.
Lucas, J. M., & Saccucci, M. S. (1990). Exponentially weighted moving average control schemes: properties and enhancements. Technometrics, 32(1), 1-12.
Marsaglia, G. (1965). Ratios of normal variables and ratios of sums of uniform variables. Journal of the American Statistical Association, 60(309), 193-204.
Malela-Majika, J. C. (2021). New distribution-free memory-type control charts based on the Wilcoxon rank-sum statistic. Quality Technology & Quantitative Management, 18(2), 135-155.
Malela-Majika, J. C., Chakraborti, S., & Graham, M. A. (2016). Distribution-free Phase II Mann–Whitney control charts with runs-rules. The International Journal of Advanced Manufacturing Technology, 86(1), 723-735.
Nguyen, H. D., Tran, K. P., & Heuchenne, C. (2019). Monitoring the ratio of two normal variables using variable sampling interval exponentially weighted moving average control charts. Quality and Reliability Engineering International, 35(1), 439-460.
ÖKsoy, D., & Aroian, L. A. (1994). The quotient of two correlated normal variables with applications. Communications in Statistics-Simulation and Computation, 23(1), 223-241.
ÖKSOY, D., Boulos, E., & DAVID PYE, L. (1993). Statistical process control by the quotient of two correlated normal variables. Quality Engineering, 6(2), 179-194.
Park, C., & Reynolds Jr, M. R. (1987). Nonparametric procedures for monitoring a location parameter based on linear placement statistics. Sequential Analysis, 6(4), 303-323.
Pignatiello Jr, J. J., & Runger, G. C. (1990). Comparisons of multivariate CUSUM charts. Journal of Quality Technology, 22(3), 173-186.
Qiu, P. (2008). Distribution-free multivariate process control based on log-linear modeling. IIE Transactions, 40(7), 664-677.
Randles, R. H. (1989). A distribution-free multivariate sign test based on interdirections. Journal of the American Statistical Association, 84(408), 1045-1050.
Randles, R. H. (2000). A simpler, affine-invariant, multivariate, distribution-free sign test. Journal of the American Statistical Association, 95(452), 1263-1268.
Roberts, S. W. (2000). Control chart tests based on geometric moving averages. Technometrics, 1(3), 239-250.
Schott, J. R. (2002). Testing for elliptical symmetry in covariance-matrix-based analyses. Statistics & Probability Letters, 60(4), 395-404.
Shewhart, W. A. (1924). Some applications of statistical methods to the analysis of physical and engineering data. Bell System Technical Journal, 3(1), 43-87.
Spisak, A. W. (1990). A control chart for ratios. Journal of Quality Technology, 22(1), 34-37.
Tran, K. P., Castagliola, P., & Celano, G. (2016). Monitoring the ratio of two normal variables using run rules type control charts. International Journal of Production Research, 54(6), 1670-1688.
Tran, K. P., Castagliola, P., & Celano, G. (2016). Monitoring the ratio of two normal variables using EWMA type control charts. Quality and Reliability Engineering International, 32(5), 1853-1869.
Tran, K. P., Castagliola, P., & Celano, G. (2018). Monitoring the ratio of population means of a bivariate normal distribution using CUSUM type control charts. Statistical Papers, 59(1), 387-413.
Tran, K. P., & Knoth, S. (2018). Steady‐state ARL analysis of ARL‐unbiased EWMA‐RZ control chart monitoring the ratio of two normal variables. Quality and Reliability Engineering International, 34(3), 377-390.
Tyler, D. E. (1987). A distribution-free M-estimator of multivariate scatter. The Annals of Statistics, 234-251.
Wang, S., & Reynolds Jr, M. R. (2013). A GLR control chart for monitoring the mean vector of a multivariate normal process. Journal of Quality Technology, 45(1), 18-33.
Yang, S. F., Lin, J. S., & Cheng, S. W. (2011). A new nonparametric EWMA sign control chart. Expert Systems with Applications, 38(5), 6239-6243.
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dc.identifier.doi (DOI) 10.6814/NCCU202201157en_US