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.

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