本研究旨在建立一個監控間隔時間的管制圖，其監控樣本服從指數分配，透過
監控間隔時間的變化來判定品質特徵是否在統計製程管制內。此管制圖可以作用在
許多領域上，例如醫療中病人到醫院就診等候的間隔時間，可以探討病人排隊等候
的時間是否增加或是感染率是否異常，也可以運用在監控火山爆發的間隔時間，判
斷火山爆發是否更加頻繁，可以讓人員做適時的避難處理。
指數累積和管制圖(cumulative sum；CUSUM)或是指數加權管制圖(exponentially
weighted moving average；EWMA)需設定管制圖之設計參數，而需設定管制圖參數
之管制圖需額外耗費時間及成本，也增加管制圖的操作難度，因此本論文設計無管
制圖參數之管制圖，讓使用者更加方便使用。本論文以資訊理論的概念建立基於指
數分配之管制圖，利用Kullback-Leibler information(KLI)來建構檢定統計量，並採
用由後往前逐期回溯樣本的方式進行檢定。後續分別探討提出之管制圖應用在參數
位移上升和參數位移下降的情況，並且透過將監控參數位移上升和監控參數位移下
降的管制圖進行結合的方式來對參數位移上升及下降同時監控，之後使用蒙地卡羅
模擬出管制圖在參數位移後的績效表現與其他指數管制圖進行比較。然而因為指數
為偏斜分配，因此會造成檢定統計量不對稱的情形，進而影響管制圖的績效表現，
因此本論文探討不同的估計手法是否會對提出之管制圖有所影響或改善。本研究管
制圖與指數CUSUM 和指數EWMA 進行比較，發現在監控參數位移上升、以及參
數位移上升及下降同時監控中，本研究提出之管制圖在整體上皆有較好的績效表現。
後續與卜瓦松KLI 管制圖的比較中，在穩定狀態下不同的參數到達率發生參數位
移上升、下降時，指數KLI 管制圖與卜瓦松KLI 管制圖各有不同的優勢。最後由
實例火山地震之間隔時間的數據中，使用本研究的管制圖與指數修華特和指數
CUSUM 進行比較，發現本研究的管制圖可以較快的偵測出變異。
關鍵字: Kullback-Leibler information、指數分配、監控間隔時間管制圖、由後往前逐
期回溯、無管制圖參數

The objective of the thesis is to build an exponential control chart for monitoring time between events. This control chart can be used in many situations including system, earthquake, medical treatment, and manufacturing process. The other control charts like EWMA and CUSUM need design parameter(s), but finding the optimal parameter(s) increases the complexity of applications. Therefore, we design a parameter-free control chart to make it more convenient. We use the concept of Kullback and Leibler information (KLI) to build an exponential KLI control chart, and use the backward sequential testing to detect the change of interested quality characteristics. Moreover, we build the control chart based on exponential distribution which has skewness. The skewness could lead to a performance of control chart decline, so we employ different methods of estimators to evaluate whether the performance would be improved or not. Finally, we use our control chart to compare with EWMA and CUSUM charts as well as a Poisson-based control chart to see which control chart is better.
From the simulation result, we found that the different estimators could improve the performance of our control chart. Depending on the different situation, the user can adopt different estimators for the exponential KLI control chart. Finally, we found that our control chart has a better performance comparing with EWMA, CUSUM control chart. But comparing with the Poisson control chart, they have different advantages in different situations.
Keywords: exponential control chart, time between events, Kullback and Leibler
information, backtracking

中文文獻:
楊瑋欣，應用幾何分佈於監控努力過程之資訊理論管制圖，國立成功大學工業與資訊
管理研究所碩士論文，民國一百零五年六月。
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