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

  • The 95% two-sided Bayesian confidence interval for [math]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]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]\beta\,\![/math] directly. Based on the information of [math]\sigma\,\![/math], we know [math]\beta_{0.005}\,\![/math] = 2 and [math]\beta_{0.995}\,\![/math] = 5. Therefore, we can use the Quick Parameter Estimator (QPE) to get the prior lognormal distribution for [math]\beta\,\![/math]. The results are Log-Mean = 1.15129 and Log-Std = 0.17786, as shown next.

WeiBays QPE.png

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

  • The 95% two-sided Bayesian confidence interval for [math]t_{0.05}\,\![/math] (B5% life) is [1623, 3452].
WeiBays B5.png
  • The 95% two-sided Bayesian confidence interval for [math]t_{0.10}\,\![/math] (B10% life) is [2030, 4763].
WeiBays B10.png
  • The 95% two-sided Bayesian confidence interval for F(2000) (probability of failure at time of 2000) is [0.014, 0.095].
WeiBays F2000.png
  • The 95% two-sided Bayesian confidence interval for F(5000) (probability of failure at time of 5000) is [0.111, 0.903].
WeiBays F5000.png

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]\sigma\,\![/math] in the book while it is for [math]\beta\,\![/math] in Weibull++.