标题: 应用无母数Bootstrap法建构Burr分布制程之平均数管制
Applying Non-parametric Bootstrap Method to Construct x-bar Control Chart for Burr Distribution process
作者: 张雅婷
唐丽英
Chang, Ya-Ting
Tong, Lee-Ing
工业工程与管理系所
关键字: bootstrap模拟法;bootstrap管制图;x-bar 管制图;Burr制程分布;bootstrap simulation method;ootstrap control chart;x-bar control chart;Bur distribution
公开日期: 2016
摘要: 利用传统x-bar管制图用来分析或管制制程之平均值时,必须假设制程资料是彼此独立且呈现常态分布;若制程分布为非常态分布时,使用传统 x ̅ 管制图可能会增加型一误差(Type I error)和型二误差(Type II error)发生的机率,使管制图失去正确侦测制程发生变异的功能。Efron在1979年提出之Bootstrap法(又称复式模拟法),可以不用对制程分布作任何统计假设,利用抽后放回之重复抽样(re-sampling)方式,以少量的样本来模拟近似母体的分布;后续又有文献提出四种Bootstrap信赖区间(SB,PB,BCPB,BCa)来估计母体参数。过去虽然有文献利用bootstrap法来建构 x ̅ 管制图之管制界限,但大多都只使用PB复式信赖区间法(Percentile Bootstrap)来建构x-bar管制图之管制界限,且没有完整比较不同复式信赖区间在非常态制程下之有效性。本研究针对同时包含左偏分布、右偏分布以及对称分布的Burr分布制程资料,利用无母数bootstrap法中的PB以及BCa两种信赖区间来建构x-bar管制图之管制界限,并利用敏感度分析来模拟验证本研究所提出之复式x-bar管制图在不同抽样样本数以及制程平均数在不同偏移程度下的有效性。本研究结果发现,当制程品质特性符合不同参数组合之Burr分布时,在制程稳定时,不论样本大小n=5或是n=2,以bootstrap法中的PB信赖区间法来建构 x-bar管制图之管制界限所得到的平均连串长度(ARL0)最佳,传统Shewart x-bar管制图次之,BCa最差;当制程失控,即平均数发生偏移时,在不同偏移程度和不同样本大小之下,应用PB以及BCa信赖区间法所建构的x-bar管制图之平均连串长度(ARL1)表现皆非常接近,且大幅优于传统Shewart x-bar管制图,其中又以BCa法略优于PB法。因此,整体而言,当制程品质特性符合不同参数组合之Burr分布(即非常态分布)且每次抽样数n介于2~5时,皆可利用本研究所建议的无母数bootstrap抽样方法以及PB信赖区间法来建构x-bar管制图之管制界限,会比传统Shewart x-bar管制图的管制效果更佳。
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.
URI: http://etd.lib.nctu.edu.tw/cdrfb3/record/nctu/#GT070353310
http://hdl.handle.net/11536/138356
显示于类别:Thesis