Appendix C: Benchmark Examples

=Reference 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:

$$\begin{matrix} {{\widehat{\sigma }}_}=0.59673 \\ \widehat{B}=9920.195 \\ \widehat{C}=9.69517\cdot {{10}^{-7}} \\ \end{matrix}$$ •

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:

$$\begin{matrix} {{\widehat{\sigma }}_}=0.59678 \\ \widehat{B}=9924.804 \\ \widehat{C}=9.58978\cdot {{10}^{-7}} \\ \end{matrix}$$

=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:

$$\begin{matrix} \widehat{\beta }=0.777 \\ \widehat{K}=6.8742\cdot {{10}^{-29}} \\ \widehat{n}=17.72958 \\ \end{matrix}$$

•	Published 95% confidence limits on $$\beta $$ :

$$\begin{matrix} \left\{ 0.653,0.923 \right\} \\ \end{matrix}$$

Computed Results for Example 2
Use the inverse power law model and Weibull as the underlying life distribution.

•	ALTA computed parameters are:

$$\begin{matrix} \widehat{\beta }=0.7765, \\ \widehat{K}=6.8741\cdot {{10}^{-29}} \\ \widehat{n}=17.7296 \\ \end{matrix}$$

•	ALTA computed 95% confidence limits on the parameters:

$$\left\{ 0.6535,0.9228 \right\}\text{ for }\widehat{\beta }$$

=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:

$$\begin{matrix} \widehat{\beta }=1.243396 \\ \widehat{K}=0.4350735 \\ \widehat{n}=13.8528 \\ \end{matrix}$$

•	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:

$$\begin{matrix} \widehat{\beta }=1.243375 \\ \widehat{K}=0.4350548 \\ \widehat{n}=13.8529 \\ \end{matrix}$$

•	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:

$$\begin{matrix} {{\widehat{\sigma }}_}=1.05, \\ \widehat{K}=1.14\cdot {{10}^{-12}}, \\ \widehat{n}=4.28. \\ \end{matrix}$$

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

Computed Results for Example 4 Use the inverse power law lognormal. •	ALTA computed parameters are:

$$\begin{matrix} {{\widehat{\sigma }}_}=1.04979 \\ \widehat{K}=1.15\cdot {{10}^{-12}} \\ \widehat{n}=4.289 \\ \end{matrix}$$

•	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:

$$\begin{matrix} \widehat{\beta }=0.4292 \\ \widehat{B}=3829.468 \\ \widehat{C}=4.513\cdot {{10}^{36}} \\ \widehat{n}=20.1 \\ \end{matrix}$$

•	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:

$$\begin{matrix} \widehat{\beta }=0.4287 \\ \widehat{B}=3780.298 \\ \widehat{C}=4.772\cdot {{10}^{36}} \\ \widehat{n}=20.09 \\ \end{matrix}$$

•	ALTA computed 95% confidence limits on the parameters: