# Weibull-Bayesian with Prior Information on Beta

 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 $\sigma\,\!$ 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.

Results in Weibull++

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++.