Weibull-Bayesian with Prior Information on Beta

This example compares the Weibull-Bayesian calculation.

The data set from Example 14.1 on page 348 in the book Statistical Methods for Reliability Data by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998 is used.

In the book, the prior distribution is set for $$\sigma\,\!$$ with $$\sigma = \frac{1}{\beta}\,\!$$. The prior for $$\sigma\,\!$$ is a lognormal distribution specified by $$\sigma_{0.005}\,\!$$ = 0.2 and $$\sigma_{0.995}\,\!$$ = 0.5. The following results are obtained using the Bayesian method:


 * The 95% two-sided Bayesian confidence interval for $$t_{0.05}\,\!$$ (B5% life) is [1613, 3236]. This result is given in Example 14.7 on page 357.
 * The 95% two-sided Bayesian confidence interval for $$t_{0.10}\,\!$$ (B10% life) is [2018, 4400]. This result is given in Example 14.7 on page 357.
 * The 95% two-sided Bayesian confidence interval for F(2000) (probability of failure at time of 2000) is [0.015, 0.097]. This result is given in Example 14.8 on page 357.
 * The 95% two-sided Bayesian confidence interval for F(5000) (probability of failure at time of 5000) is [0.132, 0.905]. This result is given in Example 14.8 on page 357.

In Weibull++, the prior distribution is set for $$\beta\,\!$$ directly. Based on the information of $$\sigma\,\!$$, we know $$\beta_{0.005}\,\!$$ = 2 and $$\beta_{0.995}\,\!$$ = 5. Therefore, we can use the Quick Parameter Estimator (QPE) to get the prior lognormal distribution for $$\beta\,\!$$. The results are Log-Mean = 1.15129 and Log-Std = 0.17786, as shown next.



Applying this prior distribution for Wei-Bayesian, we have the following results:


 * The 95% two-sided Bayesian confidence interval for $$t_{0.05}\,\!$$ (B5% life) is [1623, 3452].




 * The 95% two-sided Bayesian confidence interval for $$t_{0.10}\,\!$$ (B10% life) is [2030, 4763].




 * The 95% two-sided Bayesian confidence interval for F(2000) (probability of failure at time of 2000) is [0.014, 0.095].




 * The 95% two-sided Bayesian confidence interval for F(5000) (probability of failure at time of 5000) is [0.111, 0.903].



The results in Weibull++ are very close but not exactly the same as the results in the book. The differences are mainly caused by the fact that the prior lognormal distribution is for $$\sigma\,\!$$ in the book while it is for $$\beta\,\!$$ in Weibull++.