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題名 監控二元貝他品質變數和平均值之研究
Study of Monitoring the Mean of Sum of Bivariate Beta-Distributed Quality Variables
作者 溫怡茹
Wen, Yi-Ru
貢獻者 楊素芬
Yang, Su-Fen
溫怡茹
Wen, Yi-Ru
關鍵詞 統計製程管制
EWMA管制圖
二元貝他分配
平均連串長度
Statistical process control
EWMA control chart
Bivariate beta distribution
Average run length
日期 2025
上傳時間 1-Sep-2025 14:50:14 (UTC+8)
摘要 在品質管制的領域中，當監控的品質變數為比例時，傳統的常態假設並不適用，而Enami 等人 (2021) 也在研究中指出，在許多實務的情況中，產品品質可透過品質特性所佔比例或其總和來衡量，並且提出一種基於二元貝他分布所建立的管制圖來監控製程。然而，Enami 等人 (2021) 僅考慮樣本數為1時的情況，因此在本研究中，我們選擇延伸其方法，探討在樣本數大於1時如何監控二元貝他品質變數和的平均值。 本研究中，提出了三種監控二元貝他品質變數和平均值的管制圖。第一種為Shewhart-type D 管制圖，在不同的樣本大小下，推導出兩變數和的累積分布函數，結合蒙地卡羅模擬方法計算管制界線。第二種是標準化的指數加權移動平均 (ZEWMA-D) 管制圖，第三種是依據兩相依品質變數和是否大於其本身的期望值，進而推導指摽變數的分配來建立ZEWMA-SD管制圖。 我們以平均連串長度 (ARL) 衡量製程失控時的管制圖表現。最後，使用人類發展指數 (HDI) 資料，監控其中的兩個指標變數和之平均值是否出現異常，實務上驗證所提方法之應用性與實務價值。
In the field of statistical process control, when the monitored quality variables are proportions, the traditional normality assumptions are not suitable. Enami et al. (2021) noted that, in many practical situations, product quality can be measured by the proportion or sum of quality characteristics, and they proposed a control chart based on the bivariate beta distribution. However, their study only considered the case of a sample size of one. In this study, we extend their approach to investigate monitoring the mean of the sum of two bivariate beta-distributed quality variables with larger sample size. This study proposes three control charts for monitoring the average of the sum of bivariate Beta quality variables. The first is the Shewhart-type D chart, where the cumulative distribution function (CDF) of the sum of two variables is derived, and control limits are computed by using Monte Carlo simulation and CDF. The second is the standardized Exponentially Weighted Moving Average (ZEWMA-D) control chart. The third is the ZEWMA-SD control chart, which is developed based on the sign test to monitor the average of the sum of two bivariate beta-distributed quality variables. Control limits for the proposed three charts are determined using Monte Carlo simulation and numerical calculation. Their out-of-control (OC) detection performance is evaluated and compared using the Average Run Length (ARL) as the performance evaluation index. Finally, the application and performance of the proposed charts are demonstrated using a real-world Human Development Index (HDI) data. We use the average of the sum of two component indices in HDI data as the monitored statistic to detect abnormal changes.
參考文獻 Adamski, K., Human, S. W., & Bekker, A. (2012). A generalized multivariate beta distribution: Control charting when the measurements are from an exponential distribution. Quality and Reliability Engineering International, 28(1), 1045–1064. Bayer, F. M., Tondolo, C. M., & Müller, F. M. (2018). Beta regression control chart for monitoring fractions and proportions. Computers & Industrial Engineering, 119, 416–426. Blumenthal, D., Gumas, E. D., Shah, A., Gunja, M. Z., & Williams, R. D., II. (2024). Mirror, Mirror 2024: A portrait of the failing U.S. health system—Comparing performance in 10 nations. The Commonwealth Fund. https://doi.org/10.26099/ta0g-zp66 Enami, S., Torabi, H., & Akhavan Niaki, T. (2019). A new control chart based on a bivariate beta distribution. Journal of Statistical Research of Iran (JSRI), 16(2), 447–464. Janzen, J. H. A., & Lewis, I. M. (2025, August 23). Somalia – Civil war, conflict, famine. Encyclopaedia Britannica. https://www.britannica.com/place/Somalia/Civil-war Geneva International Centre for Humanitarian Demining. (2023, September 21). Chad – Ammunition Management Activity Platform (A-MAP). https://a-map.gichd.org/country-dashboard/chad/ Gleixner-Hayat, B. (2023). Ethiopia's fragile stability remains at risk. Carnegie Endowment for International Peace. https://carnegieendowment.org/posts/2023/11/ethiopias-fragile-stability-remains-at-risk?lang=en Ho, L. L., Fernandes, F. H., & Bourguignon, M. (2018). Control charts to monitor rates and proportions. Quality and Reliability Engineering International, 35(1), 220–235. Human Rights Watch. (2023, June 5). Malawi: Refugees, Including Children, Forcibly Relocated. https://www.hrw.org/news/2023/06/05/malawi-refugees-including-children-forcibly-relocated International Organization for Migration. (n.d.). Displacement Tracking Matrix: Uruguay. Jones, M. C. (2001). Multivariate t and beta distributions associated with the multivariate F distribution. Metrika, 54(3), 215–231. Nadarajah, S., Shih, S. H., & Nagar, D. K. (2012). A new bivariate beta distribution. Communications in Statistics – Theory and Methods, 41(6), 1065–1084. Nadarajah, S., & Kotz, S. (2005). Some bivariate beta distributions. Statistics, 39(6), 457–466. Olkin, I., & Liu, R. (2003). A bivariate beta distribution. Statistics & Probability Letters, 62(4), 407–412. Orozco-Castañeda, J. M., Nagar, D. K., & Gupta, A. K. (2012). Generalized bivariate beta distributions involving Appell’s hypergeometric function of the second kind. Computers & Mathematics with Applications, 64(8), 2507–2519. Roberts, S. W. (1959). Control chart tests based on geometric moving averages. Technometrics, 1(3), 239–250. Sant’Anna, Â. M. O., & ten Caten, C. S. (2012). Beta control charts for monitoring fraction data. Expert Systems with Applications, 39(11), 10236–10243. Sarabia, J. M., Prieto, F., & Jordá, V. (2014). Bivariate beta-generated distributions with applications to well-being data. Journal of Statistical Distributions and Applications, 1(1), 15. 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. Save the Children. (2022, December 13). Top countries where education systems most at risk of collapse: Afghanistan, Sudan, Somalia, Mali. https://www.hrw.org/news/2023/06/05/malawi-refugees-including-children-forcibly-relocated Statista. (2023). Coronavirus (COVID-19) deaths worldwide per one million population. https://www.statista.com/statistics/1104709/coronavirus-deaths-worldwide-per-million-inhabitants/ United Nations Office for Disaster Risk Reduction. (2019, December 23). United Nations Office for Disaster Risk Reduction: 2018 annual report. https://www.undrr.org/publication/united-nations-office-disaster-risk-reduction-2018-annual-report USA for UNHCR. (2025). Refugee Crisis in Europe: Aid, Statistics and News. https://www.unrefugees.org/emergencies/europe/ Worldometer. (2024). Bulgaria COVID - Coronavirus Statistics. https://www.worldometers.info/coronavirus/country/bulgaria/ Yang, S.F., & Arnold, B. C. (2016). A new approach for monitoring process variance. Journal of Statistical Computation and Simulation, 86(14), 2749–2765. https://doi.org/10.1080/00949655.2015.1125901
描述 碩士
國立政治大學
統計學系
112354023
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0112354023
資料類型 thesis
dc.contributor.advisor 楊素芬zh_TW
dc.contributor.advisor Yang, Su-Fenen_US
dc.contributor.author (Authors) 溫怡茹zh_TW
dc.contributor.author (Authors) Wen, Yi-Ruen_US
dc.creator (作者) 溫怡茹zh_TW
dc.creator (作者) Wen, Yi-Ruen_US
dc.date (日期) 2025en_US
dc.date.accessioned 1-Sep-2025 14:50:14 (UTC+8)-
dc.date.available 1-Sep-2025 14:50:14 (UTC+8)-
dc.date.issued (上傳時間) 1-Sep-2025 14:50:14 (UTC+8)-
dc.identifier (Other Identifiers) G0112354023en_US
dc.identifier.uri (URI) https://nccur.lib.nccu.edu.tw/handle/140.119/159042-
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) 112354023zh_TW
dc.description.abstract (摘要) 在品質管制的領域中，當監控的品質變數為比例時，傳統的常態假設並不適用，而Enami 等人 (2021) 也在研究中指出，在許多實務的情況中，產品品質可透過品質特性所佔比例或其總和來衡量，並且提出一種基於二元貝他分布所建立的管制圖來監控製程。然而，Enami 等人 (2021) 僅考慮樣本數為1時的情況，因此在本研究中，我們選擇延伸其方法，探討在樣本數大於1時如何監控二元貝他品質變數和的平均值。 本研究中，提出了三種監控二元貝他品質變數和平均值的管制圖。第一種為Shewhart-type D 管制圖，在不同的樣本大小下，推導出兩變數和的累積分布函數，結合蒙地卡羅模擬方法計算管制界線。第二種是標準化的指數加權移動平均 (ZEWMA-D) 管制圖，第三種是依據兩相依品質變數和是否大於其本身的期望值，進而推導指摽變數的分配來建立ZEWMA-SD管制圖。 我們以平均連串長度 (ARL) 衡量製程失控時的管制圖表現。最後，使用人類發展指數 (HDI) 資料，監控其中的兩個指標變數和之平均值是否出現異常，實務上驗證所提方法之應用性與實務價值。zh_TW
dc.description.abstract (摘要) In the field of statistical process control, when the monitored quality variables are proportions, the traditional normality assumptions are not suitable. Enami et al. (2021) noted that, in many practical situations, product quality can be measured by the proportion or sum of quality characteristics, and they proposed a control chart based on the bivariate beta distribution. However, their study only considered the case of a sample size of one. In this study, we extend their approach to investigate monitoring the mean of the sum of two bivariate beta-distributed quality variables with larger sample size. This study proposes three control charts for monitoring the average of the sum of bivariate Beta quality variables. The first is the Shewhart-type D chart, where the cumulative distribution function (CDF) of the sum of two variables is derived, and control limits are computed by using Monte Carlo simulation and CDF. The second is the standardized Exponentially Weighted Moving Average (ZEWMA-D) control chart. The third is the ZEWMA-SD control chart, which is developed based on the sign test to monitor the average of the sum of two bivariate beta-distributed quality variables. Control limits for the proposed three charts are determined using Monte Carlo simulation and numerical calculation. Their out-of-control (OC) detection performance is evaluated and compared using the Average Run Length (ARL) as the performance evaluation index. Finally, the application and performance of the proposed charts are demonstrated using a real-world Human Development Index (HDI) data. We use the average of the sum of two component indices in HDI data as the monitored statistic to detect abnormal changes.en_US
dc.description.tableofcontents 1. Introduction 11 1.1. Literature Review 11 2. The Bivariate Beta Distribution 14 2.1. Review the bivariate beta distribution 14 2.2. The distributions of estimators for the mean of the sum of two bivariate beta-distributed variables under various sample sizes 15 3. The Shewhart-type D Chart and ZEWMA-D Chart for Monitoring Mean of the Sum of the Bivariate-Beta Distributed Quality Variables 19 3.1. The Shewhart-type D chart for monitoring mean of the sum of the bivariate-beta distributed quality variables 19 3.2. Design of the ZEWMA-D chart for monitoring mean of the sum of bivariate-beta distributed quality variables 24 3.3. Detection performance of Shewhart-type D, Shewhart-type Z, ZEWMA-D and ZEWMA-D ̅ charts 28 4. The Standardized Sign-Based EWMA Control Chart for Monitoring the Mean of Sum of the Bivariate-Beta Distributed Processes 29 4.1 Design of the ZEWMA-SD chart 29 4.2 Determination of the control limits of the ZEWMA-SD chart 32 4.3 Detection performance of the ZEWMA-SD and ZEWMA-SD ̅ charts and performance comparison among the proposed control charts 33 5. A Real Example using HDI data 34 6. Conclusions 40 References 41zh_TW
dc.format.extent 3525983 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0112354023en_US
dc.subject (關鍵詞) 統計製程管制zh_TW
dc.subject (關鍵詞) EWMA管制圖zh_TW
dc.subject (關鍵詞) 二元貝他分配zh_TW
dc.subject (關鍵詞) 平均連串長度zh_TW
dc.subject (關鍵詞) Statistical process controlen_US
dc.subject (關鍵詞) EWMA control charten_US
dc.subject (關鍵詞) Bivariate beta distributionen_US
dc.subject (關鍵詞) Average run lengthen_US
dc.title (題名) 監控二元貝他品質變數和平均值之研究zh_TW
dc.title (題名) Study of Monitoring the Mean of Sum of Bivariate Beta-Distributed Quality Variablesen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) Adamski, K., Human, S. W., & Bekker, A. (2012). A generalized multivariate beta distribution: Control charting when the measurements are from an exponential distribution. Quality and Reliability Engineering International, 28(1), 1045–1064. Bayer, F. M., Tondolo, C. M., & Müller, F. M. (2018). Beta regression control chart for monitoring fractions and proportions. Computers & Industrial Engineering, 119, 416–426. Blumenthal, D., Gumas, E. D., Shah, A., Gunja, M. Z., & Williams, R. D., II. (2024). Mirror, Mirror 2024: A portrait of the failing U.S. health system—Comparing performance in 10 nations. The Commonwealth Fund. https://doi.org/10.26099/ta0g-zp66 Enami, S., Torabi, H., & Akhavan Niaki, T. (2019). A new control chart based on a bivariate beta distribution. Journal of Statistical Research of Iran (JSRI), 16(2), 447–464. Janzen, J. H. A., & Lewis, I. M. (2025, August 23). Somalia – Civil war, conflict, famine. Encyclopaedia Britannica. https://www.britannica.com/place/Somalia/Civil-war Geneva International Centre for Humanitarian Demining. (2023, September 21). Chad – Ammunition Management Activity Platform (A-MAP). https://a-map.gichd.org/country-dashboard/chad/ Gleixner-Hayat, B. (2023). Ethiopia's fragile stability remains at risk. Carnegie Endowment for International Peace. https://carnegieendowment.org/posts/2023/11/ethiopias-fragile-stability-remains-at-risk?lang=en Ho, L. L., Fernandes, F. H., & Bourguignon, M. (2018). Control charts to monitor rates and proportions. Quality and Reliability Engineering International, 35(1), 220–235. Human Rights Watch. (2023, June 5). Malawi: Refugees, Including Children, Forcibly Relocated. https://www.hrw.org/news/2023/06/05/malawi-refugees-including-children-forcibly-relocated International Organization for Migration. (n.d.). Displacement Tracking Matrix: Uruguay. Jones, M. C. (2001). Multivariate t and beta distributions associated with the multivariate F distribution. Metrika, 54(3), 215–231. Nadarajah, S., Shih, S. H., & Nagar, D. K. (2012). A new bivariate beta distribution. Communications in Statistics – Theory and Methods, 41(6), 1065–1084. Nadarajah, S., & Kotz, S. (2005). Some bivariate beta distributions. Statistics, 39(6), 457–466. Olkin, I., & Liu, R. (2003). A bivariate beta distribution. Statistics & Probability Letters, 62(4), 407–412. Orozco-Castañeda, J. M., Nagar, D. K., & Gupta, A. K. (2012). Generalized bivariate beta distributions involving Appell’s hypergeometric function of the second kind. Computers & Mathematics with Applications, 64(8), 2507–2519. Roberts, S. W. (1959). Control chart tests based on geometric moving averages. Technometrics, 1(3), 239–250. Sant’Anna, Â. M. O., & ten Caten, C. S. (2012). Beta control charts for monitoring fraction data. Expert Systems with Applications, 39(11), 10236–10243. Sarabia, J. M., Prieto, F., & Jordá, V. (2014). Bivariate beta-generated distributions with applications to well-being data. Journal of Statistical Distributions and Applications, 1(1), 15. 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. Save the Children. (2022, December 13). Top countries where education systems most at risk of collapse: Afghanistan, Sudan, Somalia, Mali. https://www.hrw.org/news/2023/06/05/malawi-refugees-including-children-forcibly-relocated Statista. (2023). Coronavirus (COVID-19) deaths worldwide per one million population. https://www.statista.com/statistics/1104709/coronavirus-deaths-worldwide-per-million-inhabitants/ United Nations Office for Disaster Risk Reduction. (2019, December 23). United Nations Office for Disaster Risk Reduction: 2018 annual report. https://www.undrr.org/publication/united-nations-office-disaster-risk-reduction-2018-annual-report USA for UNHCR. (2025). Refugee Crisis in Europe: Aid, Statistics and News. https://www.unrefugees.org/emergencies/europe/ Worldometer. (2024). Bulgaria COVID - Coronavirus Statistics. https://www.worldometers.info/coronavirus/country/bulgaria/ Yang, S.F., & Arnold, B. C. (2016). A new approach for monitoring process variance. Journal of Statistical Computation and Simulation, 86(14), 2749–2765. https://doi.org/10.1080/00949655.2015.1125901zh_TW