关于累积和(CUSUM)检验的改进  

ON THE IMPROVING OF CUMULATIVE SUM CHART

作　　者：濮晓龙[1] 

机构地区：[1]华东师范大学统计系,上海200062

出　　处：《应用数学学报》2003年第2期225-241,共17页Acta Mathematicae Applicatae Sinica

摘　　要：对连续检验问题,常用的检测方法有三大类,其一是众所周知的Shewhart控制图,它是最常用的对生产过程进行连续监控的控制方法,不过,如果过程均值有小的漂移(即μ-μ_0小)时,Shewhart控制图的检验效果不是很好,除了Shewhart控制图外,另有二类常用的控制图法,其一是累积和控制图(CUSUM),由Page基于似然比导出,其二是指数加权移动平均控制图(EWMA),由Roberts给出,它们已被证明在检测小的漂移时效果不错。许多人对CUSUM与EWMA进行了比较,总的来说,最好的CUSUM与最好的EWMA在检测小的漂移方面难分优劣,但CUSUM是由似然比导出的,且其平均运行长度的计算相对来说要简便些,因此,CUSUM在与EWMA的比较中更具优势,应用更广。我们分析了CUSUM的导出过程和公式,指出CUSUM有二个可以进一步改进的方面,在此基础上,我们给出了二个新的累积和检验统计量及其判断准则,它们分别是PCUSUM检验统计量P_n和DCUSUM检验统计量S_n.在连续检验问题中判断一个检验方法好坏的最重要的标准是其平均运行长度,比较标准是:在要求具有相同的受控状态下平均运行长度ARL_0的条件下,比较其失控状态下的平均运行长度ARL_1,ARL_1越小越好,我们对PCUSUM检验和DCUSUM检验都建立了其平均运行长度ARL的计算公式,通过对CUSUM,PCUSUM。In general, there are three methods to deal with the continuous inspection problem. One is the well-known Shewhart chart proposed by [3]?Shewhart (1924). Many authors pointed out that when the shift in mean is small (i.e. n ?u0 is small) then the Shewhart chart is not effective, the other two methods are the CUSUM (Cumulative Sum) chart proposed by PageW and the EWMA (exponential weighted moving average) chart proposed by Roberts[2]. Lots of authors compared the CUSUM with the EWMA. Generally speaking, the optimal CUSUM and the optimal EWMA both act well in detecting the small shift in mean. Since the CUSUM was laded out based the likelihood ratio and the ARL of the CUSUM is easier to compute, the CUSUM is most comparable in practice. The CUSUM test statistic was proposed by Page[1] based on the likelihood ratio. We pointed out that we can improve the CUSUM in two ways and then we give two new CUSUM test statistics and their decision principle. One is the Pn named by PCUSUM, and the other is the Sn named by DCUSUM. To judge a inspection method, one usually compute its ARL (average run length). A good test should has the smallest out-of-control ARL with the same in-control ARL. We established the formula to compute the ARL of the PCUSUM and the DCUSUM. The results of ARL show that the PCUSUM and the DCUSUM both improved the usual CUSUM.

关 键 词：连续检验问题 检测方法 累积和控制图 CUSUM 质量控制 指数加权移动平均控制图 均值漂移问题 检验统计量 平均运行长度 参数 变动和式累积和检验 变动限累积和检验 

分 类 号：O212.1[理学—概率论与数理统计] O213.1[理学—数学]