Cumulative Damage Model for Progress Stress Profiles - Revision history

Kate Racaza at 18:23, 28 September 2015

2015-09-28T18:23:19Z

← Older revision		Revision as of 18:23, 28 September 2015
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	{{Reference Example\|{{Banner ALTA Reference Examples}}}}		{{Reference Example\|{{Banner ALTA Reference Examples}}}}

	~~Compares~~ the results of a cumulative damage model for a ramp stress test.		This example validates the results of a cumulative damage model for a ramp stress test in an ALTA standard folio.

Kate Racaza at 21:51, 16 June 2014

2014-06-16T21:51:06Z

← Older revision		Revision as of 21:51, 16 June 2014
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	We get the equation for C as:		We get the equation for ''C'' as:

	::<math>C = e^{-\alpha_{0}} \cdot A^{-\alpha_{2}} \cdot R^{-\alpha_{1}} \cdot ]frac{1}{1-\alpha_{1}} \cdot t^{1-\alpha_{1}}\,\!</math>		::<math>C = e^{-\alpha_{0}} \cdot A^{-\alpha_{2}} \cdot R^{-\alpha_{1}} \cdot \frac{1}{1-\alpha_{1}} \cdot t^{1-\alpha_{1}}\,\!</math>


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	The following reliability plot shows that the predicted values (the line) by the model in ALTA can match the observed values (the points) very well.		The following reliability plot shows that the predicted values (the line) from the model in ALTA can match the observed values (the points) very well.

	[[Image: Cum Damage Stress Profile plot.png\|center\|550 px]]		[[Image: Cum Damage Stress Profile plot.png\|center\|550 px]]

Kate Racaza at 20:55, 16 June 2014

2014-06-16T20:55:21Z

← Older revision		Revision as of 20:55, 16 June 2014
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	{\| {{table}}		{\| {{table}}
	\| Time to Failure\|\|Voltage Ramp Rate\|\|Area\|\|f(t)\|\|log-f(t)		! Time to Failure\|\|Voltage Ramp Rate\|\|Area\|\|f(t)\|\|log-f(t)
	\|-		\|-
	\| 41\|\|10\|\|1\|\|0.069249\|\|-2.67004		\| 41\|\|10\|\|1\|\|0.069249\|\|-2.67004
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	The total likelihood value is LKV = -2.67004 -2.36283--2.49568…= -1035.42696.		The total likelihood value is LKV = -2.67004 -2.36283--2.49568…= -1035.42696.

	We also can use the optimization tool in Excel to find the ML solutions for this data set. By using the GRG-nonlinear optimization method provided by the Solver in Excel, the result in Excel is the same as the result in ALTA.		We also can use the optimization tool in Excel to find the ML solutions for this data set. By using the GRG-nonlinear optimization method provided by the "Solver Add-in" program in Excel, the result in Excel is the same as the result in ALTA.

			[[Image: Cum Damage_Stress Profile_excel add in.png\|center\|600 px]]


			The following reliability plot shows that the predicted values (the line) by the model in ALTA can match the observed values (the points) very well.

			[[Image: Cum Damage Stress Profile plot.png\|center\|550 px]]

Kate Racaza at 20:43, 16 June 2014

2014-06-16T20:43:16Z

← Older revision		Revision as of 20:43, 16 June 2014
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	'''Validate the Likelihood Value Calculation in ALTA'''		'''Validate the Likelihood Value Calculation in ALTA'''

	For the ML solution given in the book, the log likelihood value is -1035.4269.		For the ML solution given in the book, the log likelihood value is -1035.4269. To validate what the LK value would be if the parameter values used in the book were calculated in ALTA, we use the Alter Parameters tool, as shown next.

			[[Image: Cum Damage_Stress Profile_alter parameters.png\|center]]


			The resulting LK Value is -1489.6904, as shown next.

			[[Image: Cum Damage_Stress Profile_alter parameters results.png\|center]]


			We need to validate the calculation for the log likelihood value in ALTA to make sure it is correct.

			First, we define the life characteristic of the Weibull distribution as a function of the two stresses:

			::<math>C = \int^{t}_{0} \frac{1}{\eta(V,A)} dV = \int^{t}_{0} e^{-(\alpha_{0}+\alpha_{1} \cdot ln(R \cdot V)+\alpha_{2} \cdot ln A)} dV\,\!</math>


			We get the equation for C as:

			::<math>C = e^{-\alpha_{0}} \cdot A^{-\alpha_{2}} \cdot R^{-\alpha_{1}} \cdot ]frac{1}{1-\alpha_{1}} \cdot t^{1-\alpha_{1}}\,\!</math>


			where ''R'' is the ramping rate for the voltage.

			Therefore, the probability density function at failure time ''t'' is:

			::<math>f(t) = \beta \cdot e^{-C^{\beta}} \cdot C^{\beta} \cdot (1-\alpha_{1}) \cdot \frac{1}{t}\,\!</math>


			The following table shows the value for the pdf and its log transformation for the first several failure times. The log likelihood value for all other failure times can be calculated using the same method. The total log likelihood value is the sum of all the individual log likelihood values.

			{\| {{table}}
			\| Time to Failure\|\|Voltage Ramp Rate\|\|Area\|\|f(t)\|\|log-f(t)
			\|-
			\| 41\|\|10\|\|1\|\|0.069249\|\|-2.67004
			\|-
			\| 43\|\|10\|\|1\|\|0.094154\|\|-2.36283
			\|-
			\| 42\|\|10\|\|1\|\|0.08244\|\|-2.49568
			\|-
			\| 43\|\|10\|\|1\|\|0.094154\|\|-2.36283
			\|-
			\| 44\|\|10\|\|1\|\|0.102165\|\|-2.28117
			\|-
			\| 40\|\|10\|\|1\|\|0.05623\|\|-2.8783
			\|-
			\| 38\|\|10\|\|1\|\|0.034243\|\|-3.37427
			\|-
			\| 47\|\|10\|\|1\|\|0.083568\|\|-2.4821
			\|-
			\| 43\|\|10\|\|1\|\|0.094154\|\|-2.36283
			\|-
			\| 45\|\|10\|\|1\|\|0.104043\|\|-2.26295
			\|-
			\| …\|\|…\|\|…\|\|…\|\|…
			\|-
			\|colspan="4" style="text-align: right;"\|'''Total'''\|\|-1035.42696
			\|-
			\|}


			The total likelihood value is LKV = -2.67004 -2.36283--2.49568…= -1035.42696.

			We also can use the optimization tool in Excel to find the ML solutions for this data set. By using the GRG-nonlinear optimization method provided by the Solver in Excel, the result in Excel is the same as the result in ALTA.

Kate Racaza at 16:35, 13 June 2014

2014-06-13T16:35:26Z

← Older revision		Revision as of 16:35, 13 June 2014
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	{{Reference Example\|~~ALTA_Reference_Examples_Banner.png\|ALTA_Reference_Examples~~}}		{{Reference Example\|{{Banner ALTA Reference Examples}}}}

	Compares the results of a cumulative damage model for a ramp stress test.		Compares the results of a cumulative damage model for a ramp stress test.

Kate Racaza at 23:19, 11 June 2014

2014-06-11T23:19:09Z

← Older revision		Revision as of 23:19, 11 June 2014
Line 1:		Line 1:
	{{Reference Example\|ALTA_Reference_Examples_Banner.png\|ALTA_Reference_Examples}}		{{Reference Example\|ALTA_Reference_Examples_Banner.png\|ALTA_Reference_Examples}}

	Compares the results ~~for the~~ cumulative damage model for ramp stress.		Compares the results of a cumulative damage model for a ramp stress test.


	{{Reference_Example_Heading1}}		{{Reference_Example_Heading1}}

	The data set is from the table on page 232 in the book ''Accelerated Testing: Statistical Models, Test Plans, and Data Analysis'' by Dr. Nelson, John Wiley & Sons, 1990. The model of Eqn. (3.10) is used and the results are given in Section 3 of Table 3.2 on page 511.		The data set is from the table on page 232 in the book ''Accelerated Testing: Statistical Models, Test Plans, and Data Analysis'' by Dr. Nelson, John Wiley & Sons, 1990. The model of Eqn. (3.10) is used and the results are given in Section 3 of Table 3.2 on page 511.


	{{Reference_Example_Heading2}}		{{Reference_Example_Heading2}}
	An accelerated test employed a pair of parallel disk electrodes immersed in insulating oil. Voltage ''V'' across the pair was increased linearly with time ''t'' at a specified rate, and the voltage at oil breakdown was recorded. Since ''V = Rt'' (''R'' is the ramp rate), the time to breakdown can be recorded. The breakdown time is also affected by the two electrode areas ''A''. Three voltage linear rates and two electrode areas are used. The six stress profiles are:

			An accelerated test employed a pair of parallel disk electrodes immersed in insulating oil. The voltage ''V'' across the pair was increased linearly with time ''t'' at a specified rate, and then the voltage at oil breakdown was recorded. Since ''V = Rt'' (''R'' is the ramp rate), the time to breakdown can be recorded. The breakdown time is also affected by the two electrode areas ''A''. Three voltage linear rates and two electrode areas are used. The six stress profiles are:

	{\| {{table}}		{\| {{table}}
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	{{Reference_Example_Heading3}}		{{Reference_Example_Heading3}}

	The following life stress relationship and Weibull distribution are used ~~for~~ the data. At a constant stress ''V'', the <math>\eta\,\!</math> is:		The following life stress relationship and Weibull distribution are used to analyze the data. At a constant stress ''V'', the <math>\eta\,\!</math> is:

	::<math>ln(\eta(V, A)) = \alpha_{0}+\alpha_{1}ln(V)+\alpha_{2}ln(A)\,\!</math>		::<math>ln\left(\eta(V, A)\right) = \alpha_{0}+\alpha_{1}ln(V)+\alpha_{2}ln(A)\,\!</math>


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	When stress is varying with time, the reliability at time ''t'' is given as ~~following~~:		When stress is varying with time, the reliability at time ''t'' is given as follows:

	::<math>R(t,V,A) = e^{-\left(\int_{0}^{t}\frac{1}{\eta(x,A)}dx\right)^\beta}</math>		::<math>R(t,V,A) = e^{-\left(\int_{0}^{t}\frac{1}{\eta(x,A)}dx\right)^\beta}</math>
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	Note that the log transformation is used for voltage. At time 0, the voltage value will be 0 if the profile is defined as ''V(t) = 100*t''; therefore, to define this stress profile, we add a small number (i.e., 0.001)to the voltage function.		Note that the log transformation is used for voltage. At time 0, the voltage value will be 0 if the profile is defined as ''V(t) = 100*t''; therefore, to define this stress profile, we add a small number (i.e., 0.001) to the voltage function.

	The following picture shows the plot for this stress profile.		The following picture shows the plot for this stress profile.
Line 341:		Line 342:


	The log transformation is used for both stresses. The results in ALTA show that the calculated parameters are close to the results in the book, while the log likelihood value (~~'''~~LK Value~~'''~~)is exactly the same as the value given in the book, as shown next.		The log transformation is used for both stresses. The results in ALTA show that the calculated parameters are close to the results in the book, while the log likelihood value (LK Value) is exactly the same as the value given in the book, as shown next.

	[[Image:Cum_Damage_Stress_Profile_result1.png\|center]]		[[Image:Cum_Damage_Stress_Profile_result1.png\|center]]

Kate Racaza at 21:31, 10 June 2014

2014-06-10T21:31:09Z

← Older revision		Revision as of 21:31, 10 June 2014
Line 5:		Line 5:

	{{Reference_Example_Heading1}}		{{Reference_Example_Heading1}}
	The data set is from ~~Table~~ on page 232 in the book ''Accelerated Testing: Statistical Models, Test Plans, and Data Analysis'' by Dr. Nelson, John Wiley & Sons, 1990. ~~Model~~ of Eqn. (3.10) is used and the results are given in Section 3 of Table 3.2 on page 511.		The data set is from the table on page 232 in the book ''Accelerated Testing: Statistical Models, Test Plans, and Data Analysis'' by Dr. Nelson, John Wiley & Sons, 1990. The model of Eqn. (3.10) is used and the results are given in Section 3 of Table 3.2 on page 511.


	{{Reference_Example_Heading2}}		{{Reference_Example_Heading2}}
	An accelerated test employed a pair of parallel disk electrodes immersed in insulating oil. Voltage ''V'' across the pair was increased linearly with time ''t'' at a specified rate, and the voltage at oil breakdown was recorded. Since ''V = Rt''(''R'' is the ramp rate), the time to breakdown can be recorded. The breakdown time is also affected by the two electrode areas ''A''. Three voltage linear rates and two electrode areas are used. The six stress profiles are:		An accelerated test employed a pair of parallel disk electrodes immersed in insulating oil. Voltage ''V'' across the pair was increased linearly with time ''t'' at a specified rate, and the voltage at oil breakdown was recorded. Since ''V = Rt'' (''R'' is the ramp rate), the time to breakdown can be recorded. The breakdown time is also affected by the two electrode areas ''A''. Three voltage linear rates and two electrode areas are used. The six stress profiles are:


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	{{Reference_Example_Heading4\|ALTA}}		{{Reference_Example_Heading4\|ALTA}}

	First, we create ~~the~~ stress ~~profiles~~ in ALTA. ~~The~~ following picture shows a stress profile ~~with a ramp rate of~~ 100 volts/sec.		First, we create a stress profile in ALTA for each of three ramp rates: 10 volts/sec, 100 volts/sec and 1,000 volts/sec. For example, the following picture shows the stress profile for the 100 volts/sec ramp rate.

	[[Image:Cum_Damage_Stress_Profile_100_volts.png\|center\|550px]]		[[Image:Cum_Damage_Stress_Profile_100_volts.png\|center\|550px]]


	~~In defining this stress profile, we add a small number to the voltage function in ALTA. This is because~~ the log transformation is used for voltage. At time 0, the voltage value will be 0 if the profile is defined as ''V(t) = 100*t''.		Note that the log transformation is used for voltage. At time 0, the voltage value will be 0 if the profile is defined as ''V(t) = 100*t''; therefore, to define this stress profile, we add a small number (i.e., 0.001)to the voltage function.

	The following picture shows a plot ~~of the~~ stress profile.		The following picture shows the plot for this stress profile.

	[[Image:Cum_Damage_Stress_Profile_100_volts_plot.png\|center\|550px]]		[[Image:Cum_Damage_Stress_Profile_100_volts_plot.png\|center\|550px]]


	~~Enter~~ the ~~stress profile~~ into ~~the~~ ALTA standard folio.		The next step is to enter the failure data into an ALTA standard folio and then use the stress profiles to define the stress values, as shown next.

			[[Image:Cum_Damage_Stress_Profile_folio.png\|center\|550px]]


			The log transformation is used for both stresses. The results in ALTA show that the calculated parameters are close to the results in the book, while the log likelihood value ('''LK Value''')is exactly the same as the value given in the book, as shown next.

			[[Image:Cum_Damage_Stress_Profile_result1.png\|center]]


			'''Validate the Likelihood Value Calculation in ALTA'''

			For the ML solution given in the book, the log likelihood value is -1035.4269.

Kate Racaza at 23:01, 9 June 2014

2014-06-09T23:01:32Z

← Older revision		Revision as of 23:01, 9 June 2014
Line 322:		Line 322:


			{{Reference_Example_Heading4\|ALTA}}

	~~{{Reference_Example_Heading4~~\|ALTA}}		First, we create the stress profiles in ALTA. The following picture shows a stress profile with a ramp rate of 100 volts/sec.

			[[Image:Cum_Damage_Stress_Profile_100_volts.png\|center\|550px]]


			In defining this stress profile, we add a small number to the voltage function in ALTA. This is because the log transformation is used for voltage. At time 0, the voltage value will be 0 if the profile is defined as ''V(t) = 100*t''.

			The following picture shows a plot of the stress profile.

			[[Image:Cum_Damage_Stress_Profile_100_volts_plot.png\|center\|550px]]


			Enter the stress profile into the ALTA standard folio.

Kate Racaza at 22:51, 9 June 2014

2014-06-09T22:51:10Z

← Older revision		Revision as of 22:51, 9 June 2014
Line 5:		Line 5:

	{{Reference_Example_Heading1}}		{{Reference_Example_Heading1}}
	The data set is from Table on page 232 in the book ''Accelerated Testing: Statistical Models, Test Plans, and Data Analysis'' by Dr. Nelson, John Wiley & Sons, 1990. Model of Eqn. (3.10) is used and the results are given in Table 3.2 on page 511.		The data set is from Table on page 232 in the book ''Accelerated Testing: Statistical Models, Test Plans, and Data Analysis'' by Dr. Nelson, John Wiley & Sons, 1990. Model of Eqn. (3.10) is used and the results are given in Section 3 of Table 3.2 on page 511.


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	{{Reference_Example_Heading3}}		{{Reference_Example_Heading3}}

	The following life stress relationship and ~~the~~ Weibull distribution are used for the data. At a constant stress V, the is:		The following life stress relationship and Weibull distribution are used for the data. At a constant stress ''V'', the <math>\eta\,\!</math> is:


	~~where is the voltage and is the area.~~		::<math>ln(\eta(V, A)) = \alpha_{0}+\alpha_{1}ln(V)+\alpha_{2}ln(A)\,\!</math>

	~~The reliability function at time t and stress V is:~~


	When stress is varying with time, the reliability at time is given as following:		where ''V'' is the voltage and ''A'' is the area.

			The reliability function at time ''t'' and stress ''V'' is:

			::<math>R(t,V,A) = e^{-\left(\frac{t}{\eta(V,A)} \right)^\beta}\,\!</math>


			When stress is varying with time, the reliability at time ''t'' is given as following:

			::<math>R(t,V,A) = e^{-\left(\int_{0}^{t}\frac{1}{\eta(x,A)}dx\right)^\beta}</math>


	In the book, the following results are provided:		In the book, the following results are provided:
	• ML solution for the parameters are given in ~~section~~ 3) of Table 3.2 on page 511: ( ), , , and .
	• The maximum log likelihood is -1035.4269.		* The ML solution for the parameters are given in Section 3 of Table 3.2 on page 511: <math>1/\beta\,\!</math> = 0.07856677 (<math>\beta\,\!</math>=12.728027) ,  <math>\alpha_{0}\,\!</math> = 3.673202 ,   <math>\alpha_{1} = \,\!</math>0.05843506 ,  and  <math>\alpha_{2}=\,\!</math>-0.058626.
			* The maximum log likelihood is -1035.4269.



	{{Reference_Example_Heading4\|ALTA}}		{{Reference_Example_Heading4\|ALTA}}

Kate Racaza: Created page with '{{Reference Example|ALTA_Reference_Examples_Banner.png|ALTA_Reference_Examples}} Compares the results for the cumulative damage model for ramp stress. {{Reference_Example_Hea…'

2014-06-09T22:34:20Z

Created page with '{{Reference Example|ALTA_Reference_Examples_Banner.png|ALTA_Reference_Examples}} Compares the results for the cumulative damage model for ramp stress. {{Reference_Example_Hea…'

New page

{{Reference Example|ALTA_Reference_Examples_Banner.png|ALTA_Reference_Examples}}

Compares the results for the cumulative damage model for ramp stress.

{{Reference_Example_Heading1}}
The data set is from Table on page 232 in the book ''Accelerated Testing: Statistical Models, Test Plans, and Data Analysis'' by Dr. Nelson, John Wiley & Sons, 1990. Model of Eqn. (3.10) is used and the results are given in Table 3.2 on page 511.

{{Reference_Example_Heading2}}
An accelerated test employed a pair of parallel disk electrodes immersed in insulating oil. Voltage ''V'' across the pair was increased linearly with time ''t'' at a specified rate, and the voltage at oil breakdown was recorded. Since ''V = Rt''(''R'' is the ramp rate), the time to breakdown can be recorded. The breakdown time is also affected by the two electrode areas ''A''. Three voltage linear rates and two electrode areas are used. The six stress profiles are:

{| {{table}}
!Voltage Ramp Rate (Volts/sec)
!Area (Sq. Inch)
|-
| 10||1
|-
| 100||1
|-
| 1000||1
|-
| 10||9
|-
| 100||9
|-
| 1000||9
|-
|}

The following table shows the failure data for an area of 1 square inch.

{| {{table}}
!Time to Failure
!Voltage Ramp Rate
!Time to Failure
!Voltage Ramp Rate
!Time to Failure
!Voltage Ramp Rate
|-
| 41||10||46||100||55||1000
|-
| 43||10||50||100||57||1000
|-
| 42||10||39||100||59||1000
|-
| 43||10||36||100||57||1000
|-
| 44||10||47||100||55||1000
|-
| 40||10||55||100||60||1000
|-
| 38||10||49||100||53||1000
|-
| 47||10||58||100||51||1000
|-
| 43||10||50||100||57||1000
|-
| 45||10||48||100||54||1000
|-
| 38||10||53||100||57||1000
|-
| 44||10||54||100||64||1000
|-
| 49||10||55||100||53||1000
|-
| 42||10||37||100||63||1000
|-
| 42||10||53||100||51||1000
|-
| 51||10||52||100||62||1000
|-
| 39||10||53||100||62||1000
|-
| 34||10||50||100||56||1000
|-
| 41||10||52||100||62||1000
|-
| 41||10||50||100||57||1000
|-
| 35||10||45||100||41||1000
|-
| 44||10||48||100||41||1000
|-
| 46||10||53||100||51||1000
|-
| 39||10||50||100||58||1000
|-
| 41||10||43||100||59||1000
|-
| 40||10||50||100||60||1000
|-
| 52||10||42||100||58||1000
|-
| 40||10||45||100||55||1000
|-
| 35||10||47||100||59||1000
|-
| 40||10||34||100||63||1000
|-
| 39||10||46||100||63||1000
|-
| 46||10||42||100||53||1000
|-
| 47||10||46||100||63||1000
|-
| 44||10||46||100||61||1000
|-
| 41||10||52||100||59||1000
|-
| 46||10||47||100||53||1000
|-
| 46||10||53||100||60||1000
|-
| 42||10||52||100||58||1000
|-
| 45||10||45||100||62||1000
|-
| 42||10||47||100||56||1000
|-
| 44||10||43||100||69||1000
|-
| 41||10||45||100||65||1000
|-
| 44||10||54||100||51||1000
|-
| 38||10||51||100||56||1000
|-
| 36||10||46||100||55||1000
|-
| 44||10||55||100||57||1000
|-
| 50||10||44||100||54||1000
|-
| 47||10||49||100||63||1000
|-
| 49||10||49||100||65||1000
|-
| 46||10||53||100||65||1000
|-
| 34||10||53||100||56||1000
|-
| 47||10||54||100||54||1000
|-
| 49||10||53||100||65||1000
|-
| 43||10||53||100||60||1000
|-
| 43||10||51||100||60||1000
|-
| 48||10||48||100||64||1000
|-
| 34||10||49||100||60||1000
|-
| 38||10||52||100||54||1000
|-
| 47||10||45||100||57||1000
|-
| 35||10||49||100||61||1000
|-
|}

The following table shows the failure data for an area of 9 square inch.

{| {{table}}
!Time to Failure
!Voltage Ramp Rate
!Time to Failure
!Voltage Ramp Rate
!Time to Failure
!Voltage Ramp Rate
|-
| 33||10||43||100||50||1000
|-
| 37||10||42||100||53||1000
|-
| 38||10||45||100||50||1000
|-
| 38||10||48||100||49||1000
|-
| 38||10||38||100||53||1000
|-
| 37||10||44||100||51||1000
|-
| 27||10||37||100||47||1000
|-
| 42||10||44||100||44||1000
|-
| 39||10||43||100||53||1000
|-
| 38||10||42||100||42||1000
|-
| 42||10||43||100||49||1000
|-
| 32||10||49||100||46||1000
|-
| 42||10||44||100||50||1000
|-
| 40||10||45||100||38||1000
|-
| 32||10||50||100||48||1000
|-
| 38||10||44||100||43||1000
|-
| 36||10||44||100||52||1000
|-
| 42||10||45||100||53||1000
|-
| 20||10||41||100||52||1000
|-
| 37||10||48||100||48||1000
|-
| 43||10||45||100||45||1000
|-
| 40||10||48||100||53||1000
|-
| 38||10||43||100||52||1000
|-
| 43||10||49||100||50||1000
|-
| 39||10||50||100||55||1000
|-
| 41||10||45||100||50||1000
|-
| 35||10||45||100||43||1000
|-
| 41||10||46||100||52||1000
|-
| 40||10||47||100||50||1000
|-
| 32||10||42||100||54||1000
|-
| 38||10||47||100||51||1000
|-
| 40||10||48||100||40||1000
|-
| 37||10||47||100||52||1000
|-
| 29||10||48||100||53||1000
|-
| 31||10||39||100||47||1000
|-
| 41||10||49||100||45||1000
|-
| 38||10||44||100||53||1000
|-
| 36||10||47||100||47||1000
|-
| 35||10||34||100||54||1000
|-
| 40||10||41||100||50||1000
|-
| 37||10||45||100||32||1000
|-
| 41||10||48||100||48||1000
|-
| 36||10||44||100||53||1000
|-
| 39||10||47||100||52||1000
|-
| 43||10||45||100||45||1000
|-
| 42||10||50||100||48||1000
|-
| 43||10||40||100||48||1000
|-
| 43||10||47||100||51||1000
|-
| 41||10||47||100||53||1000
|-
| 44||10||43||100||48||1000
|-
| 37||10||49||100||54||1000
|-
| 43||10||45||100||51||1000
|-
| 38||10||45||100||50||1000
|-
| 40||10||45||100||54||1000
|-
| 40||10||47||100||35||1000
|-
| 38||10||39||100||56||1000
|-
| 33||10||44||100||51||1000
|-
| 40||10||37||100||48||1000
|-
| 35||10||47||100||48||1000
|-
| 41||10||48||100||46||1000
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The following life stress relationship and the Weibull distribution are used for the data. At a constant stress V, the is:

where is the voltage and is the area.

The reliability function at time t and stress V is:

When stress is varying with time, the reliability at time is given as following:

In the book, the following results are provided:
• ML solution for the parameters are given in section 3) of Table 3.2 on page 511: ( ), , , and .
• The maximum log likelihood is -1035.4269.

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