Appendix C: Benchmark Examples

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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:
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