Appendix C: Benchmark Examples

In this section, five published examples are presented for comparison purposes. ReliaSoft's R&D validated the Weibull++ software with hundreds of data sets and methods. Weibull++ 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 Weibull++.

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 Weibull++ by selecting the data sheet for grouped times-to-failure data with suspensions and using the Arrhenius model, the lognormal distribution, and MLE. Weibull++ 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. Weibull++ computed parameters are:

• Weibull++ 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.

• Weibull++ 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.

• Weibull++ computed parameters are:

• Weibull++ 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.

• Weibull++ computed parameters are:

• Weibull++ computed 95% confidence limits on the parameters: