應用無母數Bootstrap法建構Burr分佈製程之平均數管制

Abstract

利用傳統x-bar管制圖用來分析或管制製程之平均值時，必須假設製程資料是彼此獨立且呈現常態分佈；若製程分佈為非常態分佈時，使用傳統 x ̅ 管制圖可能會增加型一誤差(Type I error)和型二誤差(Type II error)發生的機率，使管制圖失去正確偵測製程發生變異的功能。Efron在1979年提出之Bootstrap法(又稱複式模擬法)，可以不用對製程分佈作任何統計假設，利用抽後放回之重複抽樣(re-sampling)方式，以少量的樣本來模擬近似母體的分佈；後續又有文獻提出四種Bootstrap信賴區間(SB，PB，BCPB，BCa)來估計母體參數。過去雖然有文獻利用bootstrap法來建構 x ̅ 管制圖之管制界限，但大多都只使用PB複式信賴區間法(Percentile Bootstrap)來建構x-bar管制圖之管制界限，且沒有完整比較不同複式信賴區間在非常態製程下之有效性。本研究針對同時包含左偏分佈、右偏分佈以及對稱分佈的Burr分佈製程資料，利用無母數bootstrap法中的PB以及BCa兩種信賴區間來建構x-bar管制圖之管制界限，並利用敏感度分析來模擬驗證本研究所提出之複式x-bar管制圖在不同抽樣樣本數以及製程平均數在不同偏移程度下的有效性。本研究結果發現，當製程品質特性符合不同參數組合之Burr分佈時，在製程穩定時，不論樣本大小n=5或是n=2，以bootstrap法中的PB信賴區間法來建構 x-bar管制圖之管制界限所得到的平均連串長度(ARL0)最佳，傳統Shewart x-bar管制圖次之，BCa最差；當製程失控，即平均數發生偏移時，在不同偏移程度和不同樣本大小之下，應用PB以及BCa信賴區間法所建構的x-bar管制圖之平均連串長度(ARL1)表現皆非常接近，且大幅優於傳統Shewart x-bar管制圖，其中又以BCa法略優於PB法。因此，整體而言，當製程品質特性符合不同參數組合之Burr分佈(即非常態分佈)且每次抽樣數n介於2~5時，皆可利用本研究所建議的無母數bootstrap抽樣方法以及PB信賴區間法來建構x-bar管制圖之管制界限，會比傳統Shewart x-bar管制圖的管制效果更佳。

The conventional x-bar control chart is usually utilized to analyze or monitor the process mean under the assumption that the process data are independent and follow a normal distribution. However, the probability of committing Type I and Type II errors will increase if the process data follow a non-normal distributions and the control chart will lose the ability of detecting process variation correctly. Bootstrap method is introduced by Efron in 1979. The simulated data can be obtained without any assumption of the underlying distribution. Although some studies utilized bootstrap methods to construct x-bar control limits, most of them only applied the Percentile Bootstrap (PB) confidence interval in constructing the x-bar control limits. Furthermore, they did not compare the effectiveness of four bootstrap confidence intervals in monitoring the process mean under non-normal distributions. This study utilizes two non-parametric bootstrap confidence interval (namely, PB, Bias-Corrected and Percentile Bootstrap (BCa)) to construct x-bar control chart under Burr distribution with negative skew, positive skew and symmetric distributions. The sensitivity analysis is conducted to verify the effectiveness of the proposed x-bar control chart. Some studies showed that BCa performs better than other Bootstrap confidence intervals in estimating the population parameter. However, the simulation result of this study indicates that when the data of a stable process follow a Burr distribution, the x-bar control chart constructed by the PB method has the highest average run length (ARL0), BCa method second and the traditional x-bar control chart has the poorest performance. When the process mean has small or large shift under various sample sizes, the x-bar control chart constructed by the PB and BCa methods perform closely (i.e., both have the lowest average run length (ARL1)) and outperform the traditional x-bar control chart. In summary, when the process data follow a Burr distribution and the sample size is between 2 and 5, the PB x-bar control chart is recommended.