Crow Extended Confidence Bounds - Revision history

Lisa Hacker at 20:41, 18 September 2023

2023-09-18T20:41:55Z

← Older revision		Revision as of 20:41, 18 September 2023
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	{{Template:~~RGA_BOOK_SUB~~\|Appendix D\|Crow Extended Confidence Bounds}}		{{Template:RGA_BOOK\|Appendix D\|Crow Extended Confidence Bounds}}
	In this appendix, we will present the two methods used in the RGA software to estimate the confidence bounds for the [[Crow Extended\|Crow extended]] model when applied to developmental testing data. The Fisher Matrix approach is based on the Fisher Information Matrix and is commonly employed in the reliability field. The Crow bounds were developed by Dr. Larry Crow.		In this appendix, we will present the two methods used in the RGA software to estimate the confidence bounds for the [[Crow Extended\|Crow extended]] model when applied to developmental testing data. The Fisher Matrix approach is based on the Fisher Information Matrix and is commonly employed in the reliability field. The Crow bounds were developed by Dr. Larry Crow.

David J. Groebel at 15:00, 19 May 2017

2017-05-19T15:00:43Z

← Older revision		Revision as of 15:00, 19 May 2017
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	The approximate confidence bounds on the instantaneous failure intensity are then estimated from:		The approximate confidence bounds on the instantaneous failure intensity are then estimated from:

	:<math>CB={{\hat{\lambda }}_{CA}}(T){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{CA}}(T))}/{{{\hat{\lambda }}}_{i}}(T)}}\,\!</math>		:<math>CB={{\hat{\lambda }}_{CA}}(T){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{CA}}(T))}/{{{\hat{\lambda }}}_{CA}}(T)}}\,\!</math>

	where <math>{{\lambda }_{CA}}(t)=\lambda \beta {{T}^{\beta -1}}\,\!</math>.		where <math>{{\lambda }_{CA}}(t)=\lambda \beta {{T}^{\beta -1}}\,\!</math>.

David J. Groebel at 17:14, 20 June 2014

2014-06-20T17:14:22Z

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Kate Racaza: /* Bounds on Growth Potential Failure Intensity */

2012-10-04T19:41:36Z

Bounds on Growth Potential Failure Intensity

← Older revision		Revision as of 19:41, 4 October 2012
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	<math>{{[{{\lambda }_{P}}(T)]}_{U}}\,\!</math> and <math>{{[{{\lambda }_{P}}(T)]}_{L}}\,\!</math> can be obtained from the equation given above for [[Crow_Extended_Confidence_Bounds#Bounds_on_Projected_Failure_Intensity\|Bounds on Projected Failure Intensity]].		<math>{{[{{\lambda }_{P}}(T)]}_{U}}\,\!</math> and <math>{{[{{\lambda }_{P}}(T)]}_{L}}\,\!</math> can be obtained from the equation given above for [[Crow_Extended_Confidence_Bounds#Bounds_on_Projected_Failure_Intensity\|Bounds on Projected Failure Intensity]].

	===Bounds on Growth Potential Failure Intensity===<!-- THIS SECTION HEADER IS LINKED FROM: Crow Extended. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->		===Bounds on Growth Potential Failure Intensity===<!-- THIS SECTION HEADER IS LINKED FROM: Crow Extended and another section on this page. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
	====Fisher Matrix Bounds====		====Fisher Matrix Bounds====
	If there are no BC failure modes, the growth potential failure intensity is		If there are no BC failure modes, the growth potential failure intensity is

Kate Racaza: /* Bounds on Growth Potential Failure Intensity */

2012-10-04T19:40:40Z

Bounds on Growth Potential Failure Intensity

← Older revision		Revision as of 19:40, 4 October 2012
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	===Bounds on Growth Potential Failure Intensity===<!-- THIS SECTION HEADER IS LINKED FROM: Crow Extended. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->		===Bounds on Growth Potential Failure Intensity===<!-- THIS SECTION HEADER IS LINKED FROM: Crow Extended. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
	====Fisher Matrix Bounds====		====Fisher Matrix Bounds====
	If there are no BC failure modes, the growth potential failure intensity is <math>{{\widehat{r}}_{GP}}(T)=\tfrac{{{N}_{A}}}{T}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,(1-{{d}_{i}})\tfrac{{{N}_{i}}}{T}\,\!</math>.		If there are no BC failure modes, the growth potential failure intensity is

			::<math>{{\widehat{r}}_{GP}}(T)=\tfrac{{{N}_{A}}}{T}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,(1-{{d}_{i}})\tfrac{{{N}_{i}}}{T}\,\!</math>.

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

	If there are BC failure modes, the growth potential failure intensity is <math>{{\widehat{r}}_{GP}}(T)={{\widehat{\lambda }}_{CA}}-{{\widehat{\lambda }}_{BD}}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,(1-{{d}_{i}})\tfrac{{{N}_{i}}}{T},\,\!</math> <math>Var({{\widehat{r}}_{GP}})\approx \tfrac{{{r}_{GP}}}{T}\,\!</math>. Therefore:		If there are BC failure modes, the growth potential failure intensity is

			::<math>{{\widehat{r}}_{GP}}(T)={{\widehat{\lambda }}_{CA}}-{{\widehat{\lambda }}_{BD}}+\underset{i=1}{\overset{M}{\mathop{\sum }}}\,(1-{{d}_{i}})\tfrac{{{N}_{i}}}{T},\,\!</math> <math>Var({{\widehat{r}}_{GP}})\approx \tfrac{{{r}_{GP}}}{T}\,\!</math>.

			:Therefore:

	::<math>\sqrt{T}\left( \frac{{{{\hat{r}}}_{GP}}-{{r}_{GP}}}{\sqrt{{{r}_{GP}}}} \right)\sim N(0,1)\,\!</math>		::<math>\sqrt{T}\left( \frac{{{{\hat{r}}}_{GP}}-{{r}_{GP}}}{\sqrt{{{r}_{GP}}}} \right)\sim N(0,1)\,\!</math>

Kate Racaza: /* Bounds on Projected Failure Intensity */

2012-10-04T19:39:04Z

Bounds on Projected Failure Intensity

← Older revision		Revision as of 19:39, 4 October 2012
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	===Bounds on Projected Failure Intensity===<!-- THIS SECTION HEADER IS LINKED FROM: Crow Extended. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->		===Bounds on Projected Failure Intensity===<!-- THIS SECTION HEADER IS LINKED FROM: Crow Extended. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
	====Fisher Matrix Bounds====		====Fisher Matrix Bounds====
	The projected failure intensity <math>{{\lambda }_{P}}(T)\,\!</math> must be positive, ~~thus~~ <math>\ln {{\lambda }_{P}}(T)\,\!</math> is approximately treated as being normally distributed as well:		The projected failure intensity <math>{{\lambda }_{P}}(T)\,\!</math> must be positive; thus, <math>\ln {{\lambda }_{P}}(T)\,\!</math> is approximately treated as being normally distributed as well:

	::<math>\frac{\ln {{{\hat{\lambda }}}_{P}}(T)-\ln {{\lambda }_{P}}(t)}{\sqrt{Var(\ln {{{\hat{\lambda }}}_{P}}(T)})}\sim N(0,1)\,\!</math>		::<math>\frac{\ln {{{\hat{\lambda }}}_{P}}(T)-\ln {{\lambda }_{P}}(t)}{\sqrt{Var(\ln {{{\hat{\lambda }}}_{P}}(T)})}\sim N(0,1)\,\!</math>

Kate Racaza: /* Bounds on Demonstrated Failure Intensity */

2012-10-04T19:38:04Z

Bounds on Demonstrated Failure Intensity

← Older revision		Revision as of 19:38, 4 October 2012
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	In this appendix, we will present the two methods used in the RGA software to estimate the confidence bounds for the [[Crow Extended\|Crow extended]] model when applied to developmental testing data. The Fisher Matrix approach is based on the Fisher Information Matrix and is commonly employed in the reliability field. The Crow bounds were developed by Dr. Larry Crow.		In this appendix, we will present the two methods used in the RGA software to estimate the confidence bounds for the [[Crow Extended\|Crow extended]] model when applied to developmental testing data. The Fisher Matrix approach is based on the Fisher Information Matrix and is commonly employed in the reliability field. The Crow bounds were developed by Dr. Larry Crow.

	===Bounds on Demonstrated Failure Intensity=== <!-- THIS SECTION HEADER IS LINKED FROM: Crow Extended. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->		===Bounds on Demonstrated Failure Intensity=== <!-- THIS SECTION HEADER IS LINKED FROM: Crow Extended and several sections on this page. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
	====Fisher Matrix Bounds====		====Fisher Matrix Bounds====
	If there are no BC failure modes, the demonstrated failure intensity is		If there are no BC failure modes, the demonstrated failure intensity is

Kate Racaza: /* Bounds on Demonstrated Failure Intensity */

2012-10-04T19:37:15Z

Bounds on Demonstrated Failure Intensity

← Older revision		Revision as of 19:37, 4 October 2012
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	===Bounds on Demonstrated Failure Intensity=== <!-- THIS SECTION HEADER IS LINKED FROM: Crow Extended. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->		===Bounds on Demonstrated Failure Intensity=== <!-- THIS SECTION HEADER IS LINKED FROM: Crow Extended. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
	====Fisher Matrix Bounds====		====Fisher Matrix Bounds====
	If there are no BC failure modes, the demonstrated failure intensity is <math>{{\widehat{\lambda }}_{D}}(T)=\tfrac{{{N}_{A}}+{{N}_{BD}}}{T}\,\!</math>. Thus:		If there are no BC failure modes, the demonstrated failure intensity is

			::<math>{{\widehat{\lambda }}_{D}}(T)=\tfrac{{{N}_{A}}+{{N}_{BD}}}{T}\,\!</math>.

			:Thus:

	::<math>Var({{\hat{\lambda }}_{D}}(t))=\frac{{{N}_{A}}}{{{T}^{2}}}+\frac{{{N}_{BD}}}{{{T}^{2}}}=\frac{{{\lambda }_{D}}(t)}{T}\,\!</math>		::<math>Var({{\hat{\lambda }}_{D}}(t))=\frac{{{N}_{A}}}{{{T}^{2}}}+\frac{{{N}_{BD}}}{{{T}^{2}}}=\frac{{{\lambda }_{D}}(t)}{T}\,\!</math>
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	where <math>C=\tfrac{{{z}_{1-\alpha /2}}}{\sqrt{T}}\,\!</math>.		where <math>C=\tfrac{{{z}_{1-\alpha /2}}}{\sqrt{T}}\,\!</math>.

	If there are BC failure modes, the demonstrated failure intensity, <math>{{\widehat{\lambda }}_{D}}(T)={{\widehat{\lambda }}_{CA}}\,\!</math>, is actually the instantaneous failure intensity based on all of the data. <math>{{\lambda }_{CA}}(T)\,\!</math> must be positive, ~~thus~~ <math>\ln {{\lambda }_{CA}}(T)\,\!</math> is approximately treated as being normally distributed.		If there are BC failure modes, the demonstrated failure intensity, <math>{{\widehat{\lambda }}_{D}}(T)={{\widehat{\lambda }}_{CA}}\,\!</math>, is actually the instantaneous failure intensity based on all of the data. <math>{{\lambda }_{CA}}(T)\,\!</math> must be positive; thus, <math>\ln {{\lambda }_{CA}}(T)\,\!</math> is approximately treated as being normally distributed.

	::<math>\frac{\ln {{{\hat{\lambda }}}_{CA}}(T)-\ln {{\lambda }_{CA}}(T)}{\sqrt{Var(\ln {{{\hat{\lambda }}}_{CA}}(T)})}\sim N(0,1)\,\!</math>		::<math>\frac{\ln {{{\hat{\lambda }}}_{CA}}(T)-\ln {{\lambda }_{CA}}(T)}{\sqrt{Var(\ln {{{\hat{\lambda }}}_{CA}}(T)})}\sim N(0,1)\,\!</math>
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	::<math>CB={{\hat{\lambda }}_{CA}}(T){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{CA}}(T))}/{{{\hat{\lambda }}}_{i}}(T)}}\,\!</math>		::<math>CB={{\hat{\lambda }}_{CA}}(T){{e}^{\pm {{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{CA}}(T))}/{{{\hat{\lambda }}}_{i}}(T)}}\,\!</math>


	where <math>{{\lambda }_{CA}}(t)=\lambda \beta {{T}^{\beta -1}}\,\!</math>.		where <math>{{\lambda }_{CA}}(t)=\lambda \beta {{T}^{\beta -1}}\,\!</math>.

Kate Racaza at 19:34, 4 October 2012

2012-10-04T19:34:46Z

← Older revision		Revision as of 19:34, 4 October 2012
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	{{Template:RGA_BOOK_SUB\|Appendix D\|Crow Extended Confidence Bounds}}		{{Template:RGA_BOOK_SUB\|Appendix D\|Crow Extended Confidence Bounds}}
	In this appendix, we will present the two methods used in the RGA software to estimate the confidence bounds for the [[Crow Extended]] model when applied to developmental testing data. The Fisher Matrix approach is based on the Fisher Information Matrix and is commonly employed in the reliability field. The Crow bounds were developed by Dr. Larry Crow.		In this appendix, we will present the two methods used in the RGA software to estimate the confidence bounds for the [[Crow Extended\|Crow extended]] model when applied to developmental testing data. The Fisher Matrix approach is based on the Fisher Information Matrix and is commonly employed in the reliability field. The Crow bounds were developed by Dr. Larry Crow.

	===Bounds on Demonstrated Failure Intensity=== <!-- THIS SECTION HEADER IS LINKED FROM: Crow Extended. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->		===Bounds on Demonstrated Failure Intensity=== <!-- THIS SECTION HEADER IS LINKED FROM: Crow Extended. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->

Kate Racaza at 17:05, 25 September 2012

2012-09-25T17:05:44Z

← Older revision		Revision as of 17:05, 25 September 2012
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	{{~~template~~:~~RGA BOOK~~\|D\|Crow Extended Confidence Bounds}}		{{Template:RGA_BOOK_SUB\|Appendix D\|Crow Extended Confidence Bounds}}
	In this appendix, we will present the two methods used in the RGA software to estimate the confidence bounds for the [[Crow Extended]] model when applied to developmental testing data. The Fisher Matrix approach is based on the Fisher Information Matrix and is commonly employed in the reliability field. The Crow bounds were developed by Dr. Larry Crow.		In this appendix, we will present the two methods used in the RGA software to estimate the confidence bounds for the [[Crow Extended]] model when applied to developmental testing data. The Fisher Matrix approach is based on the Fisher Information Matrix and is commonly employed in the reliability field. The Crow bounds were developed by Dr. Larry Crow.