Weibull-Bayesian with Prior Information on Beta
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Weibull-Bayesian with Prior Information on Beta |
This example validates the Weibull-Bayesian calculations in Weibull++ standard folios.
Reference Case
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.
Data
| Number in State | State F or S | Time to Failure |
|---|---|---|
| 288 | S | 50 |
| 148 | S | 150 |
| 1 | F | 230 |
| 124 | S | 250 |
| 1 | F | 334 |
| 111 | S | 350 |
| 1 | F | 423 |
| 106 | S | 450 |
| 99 | S | 550 |
| 110 | S | 650 |
| 114 | S | 750 |
| 119 | S | 850 |
| 127 | S | 950 |
| 1 | F | 990 |
| 1 | F | 1009 |
| 123 | S | 1050 |
| 93 | S | 1150 |
| 47 | S | 1250 |
| 41 | S | 1350 |
| 27 | S | 1450 |
| 1 | F | 1510 |
| 11 | S | 1550 |
| 6 | S | 1650 |
| 1 | S | 1850 |
| 2 | S | 2050 |
Result
In the book, the prior distribution is set for [math]\displaystyle{ \sigma\,\! }[/math] with [math]\displaystyle{ \sigma = \frac{1}{\beta}\,\! }[/math]. The prior for [math]\displaystyle{ \sigma\,\! }[/math] is a lognormal distribution specified by [math]\displaystyle{ \sigma_{0.005}\,\! }[/math] = 0.2 and [math]\displaystyle{ \sigma_{0.995}\,\! }[/math] = 0.5. The following results are obtained using the Bayesian method:
- The 95% two-sided Bayesian confidence interval for [math]\displaystyle{ t_{0.05}\,\! }[/math] (B5% life) is [1613, 3236]. This result is given in Example 14.7 on page 357.
- The 95% two-sided Bayesian confidence interval for [math]\displaystyle{ t_{0.10}\,\! }[/math] (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.
Results in Weibull++
In Weibull++, the prior distribution is set for [math]\displaystyle{ \beta\,\! }[/math] directly. Based on the information of [math]\displaystyle{ \sigma\,\! }[/math], we know [math]\displaystyle{ \beta_{0.005}\,\! }[/math] = 2 and [math]\displaystyle{ \beta_{0.995}\,\! }[/math] = 5. Therefore, we can use the Quick Parameter Estimator (QPE) to get the prior lognormal distribution for [math]\displaystyle{ \beta\,\! }[/math]. 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 [math]\displaystyle{ t_{0.05}\,\! }[/math] (B5% life) is [1623, 3452].
- The 95% two-sided Bayesian confidence interval for [math]\displaystyle{ t_{0.10}\,\! }[/math] (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 [math]\displaystyle{ \sigma\,\! }[/math] in the book while it is for [math]\displaystyle{ \beta\,\! }[/math] in Weibull++.