威布尔区间估计之似然比定理的适用性探讨

Discussion on applicability of the likelihood ratio theorem in terms of Weibull interval estimation

选取10台服从威布尔分布(尺度参数1500 h,形状参数1.2)的数控机床进行有替换定时截尾仿真试验(截尾时间4500 h)。然后对8个数据块(包括4种无截尾情况、4种截尾情况)进行似然比对数是否符合卡方分布的拟合优度检验。检验结果表明,5个数据块不服从卡方分布,3个数据块服从卡方分布。说明对于有替换定时截尾试验,当试验时间小于平均寿命的3倍时,应用似然比定理并取相应的极限卡方分布进行区间估计有可能是不成立的。对于较长寿命的评估对象,例如数控机床1500~2000 h寿命,工程上能实现的可靠性寿命试验时间一般不会超过平均寿命的3倍,提示对于这类产品寿命的区间估计存在似然比定理的适用性问题。

In this article,a total of 10 CNC machine tools which obeys the law of Weibull distribution(the scale parameter of 1500 h and the shape parameter of 1.2)are selected to conduct the timing-censored simulation test with replacement(the censored time of 4500 h).Then,the goodness-of-fit test is carried out on the 8 data clusters(including 4 non-censored clusters and 4 censored clusters),in order to check whether the logarithm of likelihood ratio is consistent with the chi-square distribution.The results show that the logarithm of likelihood ratio for 5 data clusters is not consistent with the chi-square distribution,and that of 3 data clusters is not consistent with the chi-square distribution.It is shown that for the timing-censored simulation test with replacement,when the test time is shorter than 3 times of the average life,perhaps it’s not valid to apply the likelihood ratio theorem and select the corresponding limit chi-square distribution for the purpose of interval estimation.For those long-life objects,such as CNC machine tools with the service life of 1500~2000 h,the test time of reliable life in terms of engineering generally does not exceed 3 times of the average life.This means that efforts should be made to check whether it’s applicable to put the likelihood ratio theorem into the interval estimation of service life for such products.