Crow-AMSAA Grouped Data Examples - Revision history

Lisa Hacker at 20:35, 18 September 2023

2023-09-18T20:35:50Z

← Older revision		Revision as of 20:35, 18 September 2023
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	<noinclude>{{Banner RGA Examples}}{{Navigation box}}		<noinclude>{{Banner RGA Examples}}{{Navigation box}}
	''These examples appear in the [~~[Crow-AMSAA (NHPP)\|~~Reliability ~~Growth and Repairable System Analysis Reference book]~~]''.		''These examples appear in the [https://help.reliasoft.com/reference/reliability_growth_and_repairable_system_analysis Reliability growth reference]''.
	</noinclude>		</noinclude>

Kate Racaza: Replaced content with '{{Banner RGA Examples}}{{Navigation box}} ''These examples appear in the Reliability Growth and Repairable System Analysis Reference book''. …'

2014-01-31T17:39:25Z

Replaced content with '<noinclude>{{Banner RGA Examples}}{{Navigation box}} ''These examples appear in the Reliability Growth and Repairable System Analysis Reference book''. …'

Show changes

Chris Kahn: /* Grouped Data - Example 2 */

2014-01-29T22:08:07Z

Grouped Data - Example 2

← Older revision		Revision as of 22:08, 29 January 2014
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	The next two figures show plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous MTBF.		The next two figures show plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous MTBF.

	[[Image:rga5.10.png\|center\|~~400px~~\|Cumulative MTBF with 2-sided 90% Fisher Matrix confidence bounds.]]		[[Image:rga5.10.png\|center\|500px\|Cumulative MTBF with 2-sided 90% Fisher Matrix confidence bounds.]]


	[[Image:rga5.11.png\|center\|~~400px~~\|Instantaneous MTBF with 2-sided 90% Fisher Matrix confidence bounds.]]		[[Image:rga5.11.png\|center\|500px\|Instantaneous MTBF with 2-sided 90% Fisher Matrix confidence bounds.]]

	The Crow confidence bounds for the cumulative and instantaneous MTBF at the 90% 2-sided confidence level and for <math>T = 500\,\!</math>hours are:		The Crow confidence bounds for the cumulative and instantaneous MTBF at the 90% 2-sided confidence level and for <math>T = 500\,\!</math>hours are:
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	The next two figures show plots of the Crow confidence bounds for the cumulative and instantaneous MTBF.		The next two figures show plots of the Crow confidence bounds for the cumulative and instantaneous MTBF.

	[[Image:rga5.12.png\|center\|~~400px~~\|Cumulative MTBF with 2-sided 90% Crow confidence bounds.]]		[[Image:rga5.12.png\|center\|500px\|Cumulative MTBF with 2-sided 90% Crow confidence bounds.]]


	[[Image:rga5.13.png\|center\|~~400px~~\|Instantaneous MTBF with 2-sided 90% Crow confidence bounds.]]		[[Image:rga5.13.png\|center\|500px\|Instantaneous MTBF with 2-sided 90% Crow confidence bounds.]]

David J. Groebel at 01:27, 25 January 2014

2014-01-25T01:27:23Z

← Older revision		Revision as of 01:27, 25 January 2014
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	2) Calculate the confidence bounds on the cumulative and instantaneous MTBF using the Fisher Matrix and Crow methods.		2) Calculate the confidence bounds on the cumulative and instantaneous MTBF using the Fisher Matrix and Crow methods.


	'''Solution'''		'''Solution'''
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	[[Image:rga5.11.png\|center\|400px\|Instantaneous MTBF with 2-sided 90% Fisher Matrix confidence bounds.]]		[[Image:rga5.11.png\|center\|400px\|Instantaneous MTBF with 2-sided 90% Fisher Matrix confidence bounds.]]


	The Crow confidence bounds for the cumulative and instantaneous MTBF at the 90% 2-sided confidence level and for <math>T = 500\,\!</math>hours are:		The Crow confidence bounds for the cumulative and instantaneous MTBF at the 90% 2-sided confidence level and for <math>T = 500\,\!</math>hours are:

David J. Groebel at 01:19, 25 January 2014

2014-01-25T01:19:23Z

← Older revision		Revision as of 01:19, 25 January 2014
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	1) Using RGA, the value of <math>\hat{\beta }\,\!</math>, must be solved numerically. Once <math>\hat{\beta }\,\!</math> has been estimated then the value of <math>\hat{\lambda }\,\!</math> can be determined. The parameter values are displayed below:		1) Using RGA, the value of <math>\hat{\beta }\,\!</math>, must be solved numerically. Once <math>\hat{\beta }\,\!</math> has been estimated then the value of <math>\hat{\lambda }\,\!</math> can be determined. The parameter values are displayed below:

	::<math>\hat{\beta }= 0.81361\,\!</math>		:<math>\hat{\beta }= 0.81361\,\!</math>

	::<math>\hat{\lambda }= 0.44585\,\!</math>		:<math>\hat{\lambda }= 0.44585\,\!</math>


	The [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29\|grouped Fisher Matrix confidence bounds]] can be obtained on the parameters <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> at the 90% confidence level by:		The [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29\|grouped Fisher Matrix confidence bounds]] can be obtained on the parameters <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> at the 90% confidence level by:

David J. Groebel at 01:14, 25 January 2014

2014-01-25T01:14:54Z

← Older revision		Revision as of 01:14, 25 January 2014
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	Using RGA, the value of <math>\hat{\beta }\,\!</math>, which must be solved numerically, is 0.6315. Using this value, the estimator of <math>\lambda \,\!</math> is:		Using RGA, the value of <math>\hat{\beta }\,\!</math>, which must be solved numerically, is 0.6315. Using this value, the estimator of <math>\lambda \,\!</math> is:

	::<math>\begin{align}		:<math>\begin{align}
	\hat{\lambda } = & \frac{11}{3,{{000}^{0.6315}}} \\		\hat{\lambda } = & \frac{11}{3,{{000}^{0.6315}}} \\
	= & 0.0701		= & 0.0701
	\end{align}\,\!</math>		\end{align}\,\!</math>


	Therefore, the intensity function becomes:		Therefore, the intensity function becomes:

	::<math>\hat{\rho }(T)=0.0701\cdot 0.6315\cdot {{T}^{-0.3685}}\,\!</math>		:<math>\hat{\rho }(T)=0.0701\cdot 0.6315\cdot {{T}^{-0.3685}}\,\!</math>

	====Grouped Data - Example 2====		====Grouped Data - Example 2====
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	The [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29\|grouped Fisher Matrix confidence bounds]] can be obtained on the parameters <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> at the 90% confidence level by:		The [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29\|grouped Fisher Matrix confidence bounds]] can be obtained on the parameters <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> at the 90% confidence level by:

	::<math>\begin{align}		:<math>\begin{align}
	{{\beta }_{L}} = & \hat{\beta }{{e}^{{{z}_{\alpha }}\sqrt{Var(\hat{\beta })}/\hat{\beta }}} \\		{{\beta }_{L}} = & \hat{\beta }{{e}^{{{z}_{\alpha }}\sqrt{Var(\hat{\beta })}/\hat{\beta }}} \\
	= & 0.6546 \\		= & 0.6546 \\
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	and:		and:

	::<math>\begin{align}		:<math>\begin{align}
	{{\lambda }_{L}} = & \hat{\lambda }{{e}^{{{z}_{\alpha }}\sqrt{Var(\hat{\lambda })}/\hat{\lambda }}} \\		{{\lambda }_{L}} = & \hat{\lambda }{{e}^{{{z}_{\alpha }}\sqrt{Var(\hat{\lambda })}/\hat{\lambda }}} \\
	= & 0.14594 \\		= & 0.14594 \\
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	= & 1.36207		= & 1.36207
	\end{align}\,\!</math>		\end{align}\,\!</math>


	Crow confidence bounds can also be obtained on the parameters <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> at the 90% confidence level, as:		Crow confidence bounds can also be obtained on the parameters <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> at the 90% confidence level, as:

	::<math>\begin{align}		:<math>\begin{align}
	{{\beta }_{L}} = & \hat{\beta }(1-S) \\		{{\beta }_{L}} = & \hat{\beta }(1-S) \\
	= & 0.63552 \\		= & 0.63552 \\
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	and:		and:

	::<math>\begin{align}		:<math>\begin{align}
	{{\lambda }_{L}} = & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot T_{k}^{\beta }} \\		{{\lambda }_{L}} = & \frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot T_{k}^{\beta }} \\
	= & 0.36197 \\		= & 0.36197 \\
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	= & 0.53697		= & 0.53697
	\end{align}\,\!</math>		\end{align}\,\!</math>


	2) The Fisher Matrix confidence bounds for the cumulative MTBF and the instantaneous MTBF at the 90% 2-sided confidence level and for <math>T=500\,\!</math> hour are:		2) The Fisher Matrix confidence bounds for the cumulative MTBF and the instantaneous MTBF at the 90% 2-sided confidence level and for <math>T=500\,\!</math> hour are:

	::<math>\begin{align}		:<math>\begin{align}
	{{[{{m}_{c}}(T)]}_{L}} = & {{{\hat{m}}}_{c}}(t){{e}^{{{z}_{\alpha /2}}\sqrt{Var({{{\hat{m}}}_{c}}(t))}/{{{\hat{m}}}_{c}}(t)}} \\		{{[{{m}_{c}}(T)]}_{L}} = & {{{\hat{m}}}_{c}}(t){{e}^{{{z}_{\alpha /2}}\sqrt{Var({{{\hat{m}}}_{c}}(t))}/{{{\hat{m}}}_{c}}(t)}} \\
	= & 5.8680 \\		= & 5.8680 \\
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	and:		and:

	::<math>\begin{align}		:<math>\begin{align}
	{{[MTB{{F}_{i}}]}_{L}} = & {{{\hat{m}}}_{i}}(t){{e}^{{{z}_{\alpha /2}}\sqrt{Var({{{\hat{m}}}_{i}}(t))}/{{{\hat{m}}}_{i}}(t)}} \\		{{[MTB{{F}_{i}}]}_{L}} = & {{{\hat{m}}}_{i}}(t){{e}^{{{z}_{\alpha /2}}\sqrt{Var({{{\hat{m}}}_{i}}(t))}/{{{\hat{m}}}_{i}}(t)}} \\
	= & 6.6483 \\		= & 6.6483 \\
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	= & 11.5932		= & 11.5932
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The next two figures show plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous MTBF.		The next two figures show plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous MTBF.
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	The Crow confidence bounds for the cumulative and instantaneous MTBF at the 90% 2-sided confidence level and for <math>T = 500\,\!</math>hours are:		The Crow confidence bounds for the cumulative and instantaneous MTBF at the 90% 2-sided confidence level and for <math>T = 500\,\!</math>hours are:

	::<math>\begin{align}		:<math>\begin{align}
	{{[{{m}_{c}}(T)]}_{L}} = & \frac{1}{C{{(t)}_{U}}} \\		{{[{{m}_{c}}(T)]}_{L}} = & \frac{1}{C{{(t)}_{U}}} \\
	= & 5.85449 \\		= & 5.85449 \\
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	and:		and:

	::<math>\begin{align}		:<math>\begin{align}
	{{[MTB{{F}_{i}}]}_{L}} = & {{\hat{m}}_{i}}(1-W) \\		{{[MTB{{F}_{i}}]}_{L}} = & {{\hat{m}}_{i}}(1-W) \\
	= & 6.19623 \\		= & 6.19623 \\
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	= & 11.36223		= & 11.36223
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The next two figures show plots of the Crow confidence bounds for the cumulative and instantaneous MTBF.		The next two figures show plots of the Crow confidence bounds for the cumulative and instantaneous MTBF.

David J. Groebel at 01:01, 25 January 2014

2014-01-25T01:01:15Z

← Older revision		Revision as of 01:01, 25 January 2014
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	Do the following:		Do the following:

	1) Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation.		1) Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation.

	2) Calculate the confidence bounds on the cumulative and instantaneous MTBF using the Fisher Matrix and Crow methods.		2) Calculate the confidence bounds on the cumulative and instantaneous MTBF using the Fisher Matrix and Crow methods.


	'''Solution'''		'''Solution'''

David J. Groebel at 01:00, 25 January 2014

2014-01-25T01:00:30Z

← Older revision		Revision as of 01:00, 25 January 2014
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	\|}		\|}

	:1) Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation.		Do the following:
	:2) Calculate the confidence bounds on the cumulative and instantaneous MTBF using the Fisher Matrix and Crow methods.		1) Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation.
			2) Calculate the confidence bounds on the cumulative and instantaneous MTBF using the Fisher Matrix and Crow methods.

	'''Solution'''		'''Solution'''

	:1) Using RGA, the value of <math>\hat{\beta }\,\!</math>, must be solved numerically. Once <math>\hat{\beta }\,\!</math> has been estimated then the value of <math>\hat{\lambda }\,\!</math> can be determined. The parameter values are displayed below:		1) Using RGA, the value of <math>\hat{\beta }\,\!</math>, must be solved numerically. Once <math>\hat{\beta }\,\!</math> has been estimated then the value of <math>\hat{\lambda }\,\!</math> can be determined. The parameter values are displayed below:

	::<math>\hat{\beta }= 0.81361\,\!</math>		::<math>\hat{\beta }= 0.81361\,\!</math>
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	:The [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29\|grouped Fisher Matrix confidence bounds]] can be obtained on the parameters <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> at the 90% confidence level by:		The [[Crow-AMSAA_Confidence_Bounds#Bounds_on_Beta_.28Grouped.29\|grouped Fisher Matrix confidence bounds]] can be obtained on the parameters <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> at the 90% confidence level by:

	::<math>\begin{align}		::<math>\begin{align}
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	:and:		and:

	::<math>\begin{align}		::<math>\begin{align}
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	:Crow confidence bounds can also be obtained on the parameters <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> at the 90% confidence level, as:		Crow confidence bounds can also be obtained on the parameters <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> at the 90% confidence level, as:

	::<math>\begin{align}		::<math>\begin{align}
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	:and:		and:

	::<math>\begin{align}		::<math>\begin{align}
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	:2) The Fisher Matrix confidence bounds for the cumulative MTBF and the instantaneous MTBF at the 90% 2-sided confidence level and for <math>T=500\,\!</math> hour are:		2) The Fisher Matrix confidence bounds for the cumulative MTBF and the instantaneous MTBF at the 90% 2-sided confidence level and for <math>T=500\,\!</math> hour are:

	::<math>\begin{align}		::<math>\begin{align}
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	:and:		and:

	::<math>\begin{align}		::<math>\begin{align}
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	:The next two figures show plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous MTBF.		The next two figures show plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous MTBF.

	[[Image:rga5.10.png\|center\|400px\|Cumulative MTBF with 2-sided 90% Fisher Matrix confidence bounds.]]		[[Image:rga5.10.png\|center\|400px\|Cumulative MTBF with 2-sided 90% Fisher Matrix confidence bounds.]]
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	:The Crow confidence bounds for the cumulative and instantaneous MTBF at the 90% 2-sided confidence level and for <math>T = 500\,\!</math>hours are:		The Crow confidence bounds for the cumulative and instantaneous MTBF at the 90% 2-sided confidence level and for <math>T = 500\,\!</math>hours are:

	::<math>\begin{align}		::<math>\begin{align}
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	:and:		and:

	::<math>\begin{align}		::<math>\begin{align}
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	:The next two figures show plots of the Crow confidence bounds for the cumulative and instantaneous MTBF.		The next two figures show plots of the Crow confidence bounds for the cumulative and instantaneous MTBF.

	[[Image:rga5.12.png\|center\|400px\|Cumulative MTBF with 2-sided 90% Crow confidence bounds.]]		[[Image:rga5.12.png\|center\|400px\|Cumulative MTBF with 2-sided 90% Crow confidence bounds.]]

David J. Groebel at 00:57, 25 January 2014

2014-01-25T00:57:24Z

← Older revision		Revision as of 00:57, 25 January 2014
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	::<math>\hat{\rho }(T)=0.0701\cdot 0.6315\cdot {{T}^{-0.3685}}\,\!</math>		::<math>\hat{\rho }(T)=0.0701\cdot 0.6315\cdot {{T}^{-0.3685}}\,\!</math>


	====Grouped Data - Example 2====		====Grouped Data - Example 2====

David J. Groebel at 00:57, 25 January 2014

2014-01-25T00:57:12Z

← Older revision		Revision as of 00:57, 25 January 2014
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	::<math>\hat{\rho }(T)=0.0701\cdot 0.6315\cdot {{T}^{-0.3685}}\,\!</math>		::<math>\hat{\rho }(T)=0.0701\cdot 0.6315\cdot {{T}^{-0.3685}}\,\!</math>