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 with . The prior for is a lognormal distribution specified by = 0.2 and = 0.5. The following results are obtained using the Bayesian method:

• The 95% two-sided Bayesian confidence interval for (B5% life) is [1613, 3236]. This result is given in Example 14.7 on page 357.
• The 95% two-sided Bayesian confidence interval for (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 directly. Based on the information of , we know = 2 and = 5. Therefore, we can use the Quick Parameter Estimator (QPE) to get the prior lognormal distribution for . 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 (B5% life) is [1623, 3452].

• The 95% two-sided Bayesian confidence interval for (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 in the book while it is for in Weibull++.