1.题目和关键词
Title:
Bayes estimation of P ( Y < X ) for the Weibull distribution with arbitrary paramets
任意参数的韦伯分布的P(Y < X)贝叶斯估计
Keywords:
Strength and stress model强度与应力模型；
Bayes credible interva贝叶斯置信区间；
Bayes estimate贝叶斯估计；
Fox-Wright function Fox-Wright函数；
MCMC method（Monte Carlo Markov Chain method，马尔科夫链蒙特卡洛方法）MCMC方法.
2.摘要
In the model of R=P(Y<X), X and Y usually represent the strength of a system and stress applied to it. Then, R is the measure of system reliability. In this paper, Bayes estimation of R=P(Y<X) is studied under the assumption that X and Y are independent Weibull random variables with arbitrary scale and shape parameters. We show here for the first time how to compute the Bayes estimates and credible intervals for R in that case. First, a closed form expression for R is derived. Prior distributions are assumed for Weibull parameters, and the posterior distribution is presented. Next, by proposing an universal sample-based method according to the Monte Carlo Markov Chain (MCMC) method, we draw samples and compute the Bayes estimates and credible intervals for R. Through Monte Carlo simulations and two real data examples, the proposed method is demonstrated to be robust and satisfactory.
在R= P（Y <X）的模型中，X和Y通常表示系统的强度和施加给它的应力。那么，R是系统可靠性的量度。本文在X和Y是具有任意尺度和形状参数的独立韦伯随机变量的假设下，研究了R = P（Y <X）的贝叶斯估计。在这里我们首次展示了在这种情况下，如何计算 R的Bayes 估计和置信区间。首先，导出R的闭式表达式（closed form expression）。先验分布被假定为韦伯参数，并提出了后验分布。接下来，通过根据马尔科夫链蒙特卡洛（MCMC）方法提出基于样本的通用方法，我们绘制样本并计算R的贝叶斯估计和置信区间。通过蒙特卡洛模拟和两个实际数据示例，所提出的方法被证明是鲁棒和令人满意的。
3.创新点、学术价值
目前还没有关于形状和尺度参数的任意值R的贝叶斯估计的研究。实际上，形状参数必须相等或比例参数必须相等是没有道理的。在某些情况下，形状参数没有明显不同和/或比例参数没有明显不同。但是，一般来说，形状和比例参数应该是自由参数。 因此，为这些参数的任意值开发贝叶斯估计程序是很重要的。这就是本文的目的。
4.对结论的理解和对学习工作的启发
Conclusion:
We have considered Bayes inference on R = P(Y < X) by assuming X and Y are absolutely different Weibull random variables. First, the expression, especially the closed form of R was derived. Assuming prior distributions for the Weibull parameters, the posterior distribution of the Weibull parameters was presented. Next, by proposing an universal sample-based method according to the MCMC method, we drew samples of the Weibull parameters and computed Bayes estimates and credible intervals for R. Through a Monte Carlo simulation study, the proposed sample-based method was shown to be robust. Finally, two real data examples were presented to show the application of the proposed method. In both the examples, the fitted results with unequal shape parameters were compared with the ones with equal shape parameters. The comparisons showed that if the true values of the shape parameters for X and Y are different, they should not be simplified to be equal as this leads to the estimate of R being different from the true value of R. Both data examples demonstrate the significance and necessity of our study.
通过假设 x 和 y 是完全不同的韦伯随机变量，我们考虑了R = P（Y <X）的贝叶斯推断。首先，导出表达式，特别是R的闭式表达式。先验分布被假定为韦伯参数，并提出了后验分布。接下来，通过MCMC方法提出一种基于通用样本的方法，我们绘制了韦伯参数的样本，并计算了R的贝叶斯估计值和可信区间。通过蒙特卡洛仿真研究，所提出的基于样本的方法被证明是可靠的。最后，给出了两个真实的数据例子来说明该方法的应用。在两个示例中，将形状参数不相等的拟合结果与形状参数相等的拟合结果进行了比较。比较表明，如果X和Y的形状参数的真实值不同，则不应将其简化为相等，因为这会导致R的估计值不同于R的真实值。这两个数据示例都证明了我们研究的重要性和必要性。
Future work:
There is still room for improvement in our study. Different priors lead to different Bayes estimates, as we saw in the simulation study and real data examples. Some future questions to address are: how to choose the appropriate prior for a specific problem? can closed form expressions be obtained for the Bayes estimates? can the study be extended if X and Y are dependent Weibull random variables?
我们的研究还有改进的空间。正如我们在模拟研究和实际数据示例中看到的那样，不同的先验值导致不同的贝叶斯估计值。未来需要解决的问题是：如何为一个特定问题选择合适的先验值？可以得到贝叶斯估计的闭式表达式吗？如果 x 和 y 是相依的威布尔随机变量，这项研究可以延伸吗？