# 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 [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.

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].

- The 95% two-sided Bayesian confidence interval for [math]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]\sigma\,\![/math] in the book while it is for [math]\beta\,\![/math] in Weibull++.