逆威布尔部件的可靠性估计
Estimating Reliability of Inverse Weibull Component

针对贝叶斯分析中平方误差损失存在的"高估和低估同等重要"问题,提出了一种基于熵损失函数的贝叶斯可靠性分析方法。利用该方法,分别在无信息先验和共轭先验分布下,推导出逆威布尔部件参数、可靠度函数及失效率的Bayes估计,并证明了形如[cT(x)+d]-1的一类估计具有容许性。为了比较不同估计结果的忧劣,文中还给出了逆威布尔部件参数的一致最小方差无偏估计(UMVUE)。最后运用Monte Carlo方法对各种估计的均方误差进行了模拟比较。结果表明,当样本量比较小时,Bayes估计的均方误差小于UMVUE的均方误差。随着样本量的增加,各个估计的均方误差都减小,但在共轭先验下Bayes估计的均方误差最小。
The mean square error loss in the Bayes estimation has the problem of "equal importance of overestimation and underestimation". Hence we propose the Bayes reliability analysis method based on the entropy loss function. With this method, we derive respectively the parameters, reliability function and failure rate function of the inverse Weibull component under non-informative priori distribution and conjugate priori distribution. We also prove that the estimation of the class [cT(x)+ d]-1 has admissibility. In order to compare the advantages and disadvantages of different estimation results, we derive the uniform minimum variance unbiased estimate (UMVUE) of the parameters of the inverse Weibull component. Finally, we use the Monte Carlo method to carry out the calculation of the mean square errors of various estimations to analyze the influence of different sample sizes on the accuracy of different estimation results and to compare the effects of the Bayes estimation under non-informative priori distribution and conjunctional prior distribution respectively. The calculation results, given in Table 1, and their analysis show preliminarily that: (1) when the sample size is relatively small, the mean square error of the Bayes estimation is smaller than that of UMVUE; (2) the mean square error of each estimation decreases with increasing sample size; (3) under conjugate priori distribution, the Bayes estimation has minimum mean square error