監控服從韋伯分佈的壽命在品質管制中是一個重要的議題。傳統上，在第二階段線上監控時，常假設管制狀態下的參數為已知，但事實上是用大量第一階段管制狀態的資料來估計。在本文中，利用標準化後的概似比檢定統計量，我們提出兩種EWMA管制圖，在假設韋伯形狀參數已知且不變的情況下監控尺度參數。在管制狀態下的尺度參數值未知，且僅有少量第一階段資料時，這兩種EWMA管制圖皆有不錯的表現。並且，這兩個管制圖還能在發出失控警訊的同時，估計改變點的位置，以及管制狀態和失控狀態下的參數。統計模擬的結果顯示，我們提出的EWMA管制圖在大部分尺度參數改變的情境下皆優於Dogu和Noor-ul-Amin (2023) 提出的自啟動EWMA管制圖。此外，在壽命實驗中，資料時常以設限(censored)的形式被收集。因此，我們將提出的管制圖推廣到韋伯分佈的型二設限(Type II censored)資料上。最後，我們透過一個實際的例子來說明如何使用本文所提出的管制圖。

Monitoring the lifetime of Weibull distribution is a crucial task in the field of statistical quality control. Traditionally, the in-control parameters are often assumed to be known on Phase II online monitoring of Weibull process. In fact, a large amount of Phase I in-control data are required to estimate them. In this article, based on the standardized likelihood ratio statistic, we propose two types of exponentially weighted moving average (EWMA) chart to monitor the Weibull scale parameter (or lifetime) under the assumption that the shape parameter is fixed and known. The proposed charts perform well without knowing the in-control scale parameter with only a few Phase I in-control data. Moreover, the proposed charting schemes can estimate the unknown change-point and the in-control and out-of-control parameters at the same time when the proposed charts trigger a signal. Simulation results show that the proposed charts outperform the self-starting EWMA chart of Dogu and Noor-ul-Amin (2023) in most of the out-of-control scenarios considered. Furthermore, we can extend our charting methods to Weibull Type II censored data to accommodate the censoring mechanism in lifetime experiments. Finally, we use an example to demonstrate the applicability of the proposed charts.

1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Monitoring lifetime data . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Purpose and motivation . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Control charts with limited Phase I data 5
2.1 A self-starting EWMA chart for exponential distribution . . . . . . . 5
2.2 Two EWMA charts based on likelihood ratio test statistic . . . . . . 7
2.2.1 Row-wise EWMA chart . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Column-wise EWMA chart . . . . . . . . . . . . . . . . . . . 12
3 Performance comparisons 14
3.1 Control limits of the R-EWMA and C-EWMA charts . . . . . . . . . 14
3.2 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.1 Comparisons of ARL . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.2 Performance of the change-point estimation . . . . . . . . . . 20
3.3 An illustrative example . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Extensions and conclusions 24
4.1 Extension to Type II censored data . . . . . . . . . . . . . . . . . . . 24
4.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
References 28
Tables 31
Figures 44

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