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Weibull-Bayesian with Prior Information on Beta
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This example compares the Weibull-Bayesian calculation.
Reference Case
The data 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 get the prior lognormal distribution for [math]\displaystyle{ \beta\,\! }[/math]. It is Log-Mean = 1.15129 and Log-Std = 0.17786.
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++.
