# Appendix C: Benchmark Examples

### From ReliaWiki

In this section, five published examples are presented for comparison purposes. ReliaSoft's R&D validated the ALTA software with hundreds of data sets and methods. ALTA also cross-validates each provided solution by independently re-evaluating the second partial derivatives based on the estimated parameters each time a calculation is performed. These partials will be equal to zero when a solution is reached. Double precision is used throughout ALTA.

# Example 1

From Wayne Nelson [28, p. 135].

## Published Results for Example 1

- Published Results:

## Computed Results for Example 1

This same data set can be entered into ALTA by selecting the data sheet for grouped times-to-failure data with suspensions and using the Arrhenius model, the lognormal distribution, and MLE. ALTA computed parameters for maximum likelihood are:

# Example 2

From Wayne Nelson [28, p. 453], time to breakdown of a transformer oil, tested at 26kV, 28kV, 30kV, 32kV, 34kV, 36kV and 38kV.

## Published Results for Example 2

- Published Results:

- Published 95% confidence limits on :

## Computed Results for Example 2

Use the inverse power law model and Weibull as the underlying life distribution. ALTA computed parameters are:

- ALTA computed 95% confidence limits on the parameters:

# Example 3

From Wayne Nelson [28, p. 157], forty bearings were tested to failure at four different test loads. The data were analyzed using the inverse power law Weibull model.

## Published Results for Example 3

Nelson's [28, p. 306] IPL-Weibull parameter estimates:

- The 95% 2-sided confidence bounds on the parameters:

- Percentile estimates at a stress of 0.87, with 95% 2-sided confidence bounds:

Percentile | Life Estimate | 95% Lower | 95% Upper |
---|---|---|---|

1% | 0.3913096 | 0.1251383 | 1.223632 |

10% | 2.589731 | 1.230454 | 5.450596 |

90% | 30.94404 | 19.41020 | 49.33149 |

99% | 54.03563 | 33.02691 | 88.40821 |

## Computed Results for Example 3

Use the inverse power law model and Weibull as the underlying life distribution.

- ALTA computed parameters are:

- The 95% 2-sided confidence bounds on the parameters:

- Percentile estimates at a stress of 0.87, with 95% 2-sided confidence bounds:

Percentile | Life Estimate | 95% Lower | 95% Upper |
---|---|---|---|

1% | 0.3913095 | 0.1251097 | 1.223911 |

10% | 2.589814 | 1.230384 | 5.451588 |

90% | 30.94632 | 19.40876 | 49.34240 |

99% | 54.04012 | 33.02411 | 88.43039 |

# Example 4

From Meeker and Escobar [26, p. 504], Mylar-Polyurethane Insulating Structure data using the inverse power law lognormal model.

## Published Results for Example 4

- Published Results:

- The 95% 2-sided confidence bounds on the parameters:

## Computed Results for Example 4

Use the inverse power law lognormal.

- ALTA computed parameters are:

- ALTA computed 95% confidence limits on the parameters:

# Example 5

From Meeker and Escobar [26, p. 515], Tantalum capacitor data using the combination (Temperature-NonThermal) Weibull model.

## Published Results for Example 5

- Published Results:

- The 95% 2-sided confidence bounds on the parameters:

## Computed Results for Example 5

Use the Temperature-NonThermal model and Weibull as the underlying life distribution.

- ALTA computed parameters are:

- ALTA computed 95% confidence limits on the parameters: