带参数估计的极差控制图设计

现实中,过程参数常常未知,需由第Ⅰ阶段的受控样本数据估计得到.不同的第Ⅰ阶段样本数据集对应着目标参数的不同估计值,进而会导致不同的控制限与不同的控制图表现.对于某位实际工作人员而言,最可能的情况是他手里仅有一组第Ⅰ阶段数据集,因此研究在给定一组第Ⅰ阶段数据集下控制图的表现,即条件表现,更具实际意义.基于Monter Carlo模拟,研究了基于样本平均极差,样本平均标准差和样本合并标准差等3种参数估计形式下常见的等尾极差图和无偏极差图的条件平均链长分布,结果表明参数估计对控制图影响严重.为了弥补第Ⅰ阶段数据量的不足,基于bootstrap方法,提出了修正控制图以获得理想的条件受控表现.比较结果显示,基于样本合并标准差的估计方法更好,修正的无偏极差图表现优于相应的修正等尾极差图.

The designs of the R control charts with estimated parameters

In practical applications,the in-control process parameters are often unknown and have to be estimated from an in-control Phase I data set.Therefore,the performance of Phase II control charts depends on the control limits established on the basis of the estimates of the in-control parameters.Different Phase I data sets can result in different parameter estimates,therefore resulting in different control limits and control charts performance.However,mostly one practitioner only has one Phase I data set,so it is necessary and meaningful to study the performance of control charts based on a given Phase I data set,i.e.,the conditional performance.Using Monte Carlo simulations,this paper considers the conditional performance of the equal-tailed and unbiased R control charts with three kinds of parameter estimates,the average sample range,the average sample standard deviation and the pooled sample standard deviation,for monitoring changes in the standard deviation of a normal process.This paper also evaluates the performance of each R chart in terms of the percentile,mean and standard deviation of conditional in-control ARL distribution.The results show that parameter estimation has a significant effect on the control charts performance,resulting in more frequent false alarms than expected.In order to solve this question,this paper proposes a bootstrap method to ad.just the control limits to obtain the control charts with desired conditional in-control performance.Results show that the third estimate of the process standard deviation is better than the first two estimates.In addition,the proposed unbiased R chart with adjusted limits outperforms than the corresponding equal-tailed chart in terms of the conditional ARL distribution.