关于累积和(CUSUM)检验的改进

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Shewhartt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-μo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^1]基于似然比导出，其二是指数加权移动平均控制图(EWMA)，由Roberts^2]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难则，它们分别是PCUSUM检验统计量Pn和DCUSUM检验统计量Sn．在连续检验问题中判断一个检验方法好坏的最重要的标难是其平均运行长度比较标难是在要求具有相同的受控状态下平均运行长度ARL0的条件下，比较其失控状态下的平均运行长度ARL1，ARL1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，DCUSUM的平均运行长度的比较我们发现我们提出的新的累积和控制方法确比原来的CUSUM有较大改进。

ON THE IMPROVING OF CUMULATIVE SUM CHART

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 Roberts2]. 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 Page1] 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.