Appendix C: Benchmark Examples

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:

[math]\displaystyle{ \begin{align} {{\widehat{\sigma }}_{{{T}'}}}= & 0.59673 \\ \widehat{B}= & 9920.195 \\ \widehat{C}= & 9.69517\cdot {{10}^{-7}} \\ \end{align} }[/math]

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:

[math]\displaystyle{ \begin{matrix} {{\widehat{\sigma }}_{{{T}'}}}=0.59678 \\ \widehat{B}=9924.804 \\ \widehat{C}=9.58978\cdot {{10}^{-7}} \\ \end{matrix} }[/math]

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:

[math]\displaystyle{ \begin{matrix} \widehat{\beta }=0.777 \\ \widehat{K}=6.8742\cdot {{10}^{-29}} \\ \widehat{n}=17.72958 \\ \end{matrix} }[/math]

• Published 95% confidence limits on [math]\displaystyle{ \beta }[/math] :

[math]\displaystyle{ \begin{matrix} \left\{ 0.653,0.923 \right\} \\ \end{matrix} }[/math]

Computed Results for Example 2

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

[math]\displaystyle{ \begin{matrix} \widehat{\beta }=0.7765, \\ \widehat{K}=6.8741\cdot {{10}^{-29}} \\ \widehat{n}=17.7296 \\ \end{matrix} }[/math]

• ALTA computed 95% confidence limits on the parameters:

[math]\displaystyle{ \left\{ 0.6535,0.9228 \right\}\text{ for }\widehat{\beta } }[/math]

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:

[math]\displaystyle{ \begin{matrix} \widehat{\beta }=1.243396 \\ \widehat{K}=0.4350735 \\ \widehat{n}=13.8528 \\ \end{matrix} }[/math]

• 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:

[math]\displaystyle{ \begin{matrix} \widehat{\beta }=1.243375 \\ \widehat{K}=0.4350548 \\ \widehat{n}=13.8529 \\ \end{matrix} }[/math]

• 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:

[math]\displaystyle{ \begin{matrix} {{\widehat{\sigma }}_{{{T}'}}}=1.05, \\ \widehat{K}=1.14\cdot {{10}^{-12}}, \\ \widehat{n}=4.28. \\ \end{matrix} }[/math]

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

Computed Results for Example 4

Use the inverse power law lognormal.

• ALTA computed parameters are:

[math]\displaystyle{ \begin{matrix} {{\widehat{\sigma }}_{{{T}'}}}=1.04979 \\ \widehat{K}=1.15\cdot {{10}^{-12}} \\ \widehat{n}=4.289 \\ \end{matrix} }[/math]

• 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:

[math]\displaystyle{ \begin{matrix} \widehat{\beta }=0.4292 \\ \widehat{B}=3829.468 \\ \widehat{C}=4.513\cdot {{10}^{36}} \\ \widehat{n}=20.1 \\ \end{matrix} }[/math]

• 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:

[math]\displaystyle{ \begin{matrix} \widehat{\beta }=0.4287 \\ \widehat{B}=3780.298 \\ \widehat{C}=4.772\cdot {{10}^{36}} \\ \widehat{n}=20.09 \\ \end{matrix} }[/math]

• ALTA computed 95% confidence limits on the parameters: