近年來，已有多位學者鑑於傳統的Hotelling’s T^2管制圖在第一階段(Phase I, trial period)無法有效偵測多變量製程中非隨機變異的樣式(non-random patterns)如趨勢(trend)、製程平均移動(process mean shift)或出現離群值(outliers)等製程失控狀態，而嘗試以穩健估計量(robust estimators)的方式建立T^2管制圖。Holmes與Mergen (1993)、Sullivan與Woodall(1996b)以樣本平均數及平均均方連續差建立多變量管制圖，有效地偵測趨勢、製程平均移動，但對離群值之偵測效果卻較差。Vargas(2003)所提出之最小體積橢圓(minimum volume ellipsoid, MVE)固然可有效偵測離群值，但對趨勢、製程平均移動等偵測效果反較差。因此，本研究計畫擬結合上述兩種穩健估計量的優點提出一個新的穩健估計量，並將以模擬的方式估算當多變量製程出現不同非隨機變異的情況下新穩健估計量所建立T^2管制圖的偵測機率(signal probability)。在本論文的第三章，我們透過多個數值實例說明新穩健估計量所建立之T^2管制圖(T^2_{WD}管制圖)可有效地偵測出多變量製程所產生的各種非隨機變異，且在偵測多變量製程可能同時出現趨勢、離群值、製程平均移動等異常變動時有其優越性。
為了進一步評估以穩健估計量建立的多變量管制圖對第二階段(Phase II, monitoring period)偵測能力的影響，在本論文的第四章我們提出衡量第二階段多變量管制圖的方法。並以箱形圖呈現偵測能力模擬結果之分佈情形，進而使用條件平均連串長度(conditional average run length)之期望值及條件連串長度標準差(conditional standard deviation of run length)之中位數作為衡量多變量管制圖長期偵測能力之評估值。藉由多變量管制圖偵測能力的比較結果得知，以新穩健估計量所建立之T^2_{WD}管制圖亦可有效地偵測出第二階段多變量製程之失控狀態。最後，我們以數個數值實例說明製程參數估計量對第二階段多變量管制圖偵測能力的影響。

In the past decade, different robust estimators have been proposed by several researchers to improve the ability for detecting non-random patterns such as trend, process mean shift, and outliers in multivariate Phase I control charts. Though the sample mean vector and the mean square successive difference matrix in the T^2 control chart (Holmes and Mergern, 1993; Sullivan and Woodall, 1996b) is sensitive to the detection of process mean shifts or trends, it is less sensitive to the detection of outliers. Conversely, the minimum volume ellipsoid (MVE) estimators in the T^2 control chart (Vargas, 2003) are sensitive to the detection of outliers, but less sensitive to the detection of trends or shifts in the process mean. Hence, we propose new robust estimators that use the merits of both the mean square successive difference matrix and the MVE estimators in Hotelling’s T^2 control chart. To compare the detection performance of various control charts, a simulation approach has been adopted to estimate control limits and signal probabilities. Our simulation results show that T^2 control chart using the new robust estimators (T^2_{WD} control chart) achieve a well-balanced sensitivity when detecting non-random patterns. In the first part of this dissertation, we demonstrate the usefulness and robustness of our new estimators using three numerical examples.
Since such estimators have not been studied before in a multivariate Phase II control chart, the second part of this dissertation proposes an evaluation method for measuring and comparing the detection performances of various T^2 Phase II control charts. We use box-plots to illustrate our simulation results. The expected value of the conditional average run length (ARL) and the median of the conditional standard deviation of run length (SDRL) are used to evaluate long-run detection performance. The simulation results also indicate that the proposed T^2_{WD} control chart using our new robust estimators achieve well-balanced detection performance in Phase II. These effects are then demonstrated in three numerical examples.

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