隨著科技的日新月異，對於產品品質的要求也隨之提升，而產品壽命的長短往往反映著產品品質的好壞，因此監控產品的壽命變化是一個重要的議題。在現今工業界裡，韋伯分佈常被用來描述產品的失效時間。在既有的文獻中，已有許多研究提出有效監控韋伯分佈參數的各種管制圖。在本文中，我們主要是利用概似比檢定建構一種監控韋伯分佈形狀參數的改變點偵測管制圖。接著我們引入指數加權移動平均的機制來改進改變點偵測管制圖的監控效率，並介紹一種能快速計算韋伯分佈形狀參數最大概似估計值的方法，藉此來更快速地尋找相關的管制界線。同時，我們將提出的管制圖與文獻上現有的一種監控方法進行比較。最後我們將提出的監控方法應用於一筆碳纖維拉伸力度資料來說明所提出的改變點偵測管制圖如何在實務上執行與運用。

With rapid advancement of technology, there has been an increased demand for product quality. The lifespan of a product often serves as a reflection of its quality, and thus it highlights the importance of monitoring product lifespans. In the contemporary industrial sector, the Weibull distribution is frequently employed to describe product lifetimes. Numerous control charts have been proposed in recent literature to effectively monitor the parameters of the Weibull distribution. In this study, our primary focus is on developing a control chart that utilizes likelihood ratio tests to detect changes in the shape parameter of the Weibull distribution. Additionally, we incorporate the mechanism of exponentially weighted moving average to enhance the monitoring efficiency of the proposed change-point detection control chart. We also present a method for rapidly calculating the maximum likelihood estimate of the shape parameter, which enables quicker determination of the relevant control limits. To evaluate the effectiveness of our proposed control chart, we compare it with an existing monitoring scheme developed in the literature. Furthermore, we apply the proposed control charts to a dataset of breaking strength of carbon fiber process to demonstrate its implementation and applicability.

目錄
第一章 緒論 1
1.1 管制圖簡介 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 監控韋伯 (Weibull) 分佈形狀參數的管制圖 . . . . . . . . . . . . . . 2
1.3 研究目的與動機 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
第二章 韋伯分佈形狀參數的監控 5
2.1 監控形狀參數的 chi-EWMA 管制圖 . . . . . . . . . . . . . . . . . . 5
2.2 利用概似比檢定的改變點偵測管制圖 . . . . . . . . . . . . . . . . . . 7
2.3 EWMA 機制的改變點偵測管制圖 . . . . . . . . . . . . . . . . . . . . 10
2.3.1 固定 n 做 EWMA . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.2 固定 r 做 EWMA . . . . . . . . . . . . . . . . . . . . . . . . 11
第三章 管制圖的監控效率比較 13
3.1 管制界限 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 管制圖效率比較 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 製程改變點的估計 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
第四章 實例 21
第五章 結論與後續研究 23
參考文獻 25
附表 28
附圖 52

Akhundjanov, S. B. and Pascual, F. (2015). Moving range EWMA control charts for
monitoring the Weibull shape parameter. Journal of Statistical Computation and
Simulation, 85(9):1864–1882.
Chan, Y., Han, B., and Pascual, F. (2015). Monitoring the Weibull shape parameter
with Type-II censored data. Quality and Reliability Engineering International,
31(5):741–760.
Guo, B. and Wang, B. X. (2014). Control charts for monitoring the Weibull shape
parameter based on Type-II censored sample. Quality and Reliability Engineering
International, 30(1):13–24.
Hawkins, D. M. (1987). Self-starting cusum charts for location and scale. Journal of
the Royal Statistical Society: Series D (The Statistician), 36(4):299–316.
Hawkins, D. M., Qiu, P., and Kang, C. W. (2003). The changepoint model for
statistical process control. Journal of Quality Technology, 35(4):355–366.
Herd, G. R. (1960). Estimation of reliability from incomplete data. In Proceedings of
the 6th national symposium on reliability and quality control, pages 202–217. IEEE
New York.
Huwang, L. and Lin, L.-W. (2020). New EWMA control charts for monitoring
the Weibull shape parameter. Quality and Reliability Engineering International,
36(6):1872–1894.
Johnson, L. G. (1964). The Statistical Treatment of Fatigue Experiments. Elsevier,
Amsterdam.
Kim, N. (2016). On the maximum likelihood estimators for parameters of a Weibull
distribution under random censoring. Communications for Statistical Applications
and Methods, 23:241–250.
Meeker, W. Q. and Escobar, L. A. (1998). Statistical Methods for Reliability Data.
John Wiley and Sons Inc.
Padgett, W. J. and Spurrier, J. D. (1990). Shewhart-type charts for percentiles of
strength distributions. Journal of Quality Technology, 22(4):283–288.
Page, E. S. (1954). Continuous inspection schemes. Biometrika, 41(1/2):100–115.
Pascual, F. (2010). EWMA charts for the Weibull shape parameter. Journal of
Quality Technology, 42(4):400–416.
Pascual, F. and Li, S. (2012). Monitoring the Weibull shape parameter by control
charts for the sample range of Type-II censored data. Quality and Reliability En-
gineering International, 28(2):233–246.
Roberts, S. W. (1959). Control chart tests based on geometric moving averages.
Technometrics, 1(3):239–250.
Saccucci, M. S. and Lucas, J. M. (1990). Average run lengths for exponentially
weighted moving average control schemes using the Markov chain approach. Jour-
nal of Quality Technology, 22(2):154–162.
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.
Zhang, C. W., Ye, Z., and Xie, M. (2017). Monitoring the shape parameter of a
Weibull renewal process. IISE Transactions, 49(8):800–813.
Zou, C., Zhang, Y., and Wang, Z. (2006). A control chart based on a change-point
model for monitoring linear profiles. IIE Transactions, 38(12):1093–1103.