Crow-AMSAA Confidence Bounds - Revision history

Lisa Hacker at 20:42, 18 September 2023

2023-09-18T20:42:05Z

← Older revision		Revision as of 20:42, 18 September 2023
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	{{Template:~~RGA_BOOK_SUB~~\|Appendix C\|Crow-AMSAA Confidence Bounds}}		{{Template:RGA_BOOK\|Appendix C\|Crow-AMSAA Confidence Bounds}}
	In this appendix, we will present the two methods used in the RGA software to estimate the confidence bounds for the [[Crow-AMSAA (NHPP)\|Crow-AMSAA (NHPP)]] 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-AMSAA (NHPP)\|Crow-AMSAA (NHPP)]] 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.

Chris Kahn at 21:25, 23 April 2014

2014-04-23T21:25:23Z

← Older revision		Revision as of 21:25, 23 April 2014
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	{{Template:RGA_BOOK_SUB\|Appendix C\|Crow-AMSAA Confidence Bounds}}		{{Template:RGA_BOOK_SUB\|Appendix C\|Crow-AMSAA Confidence Bounds}}
	In this appendix, we will present the two methods used in the RGA software to estimate the confidence bounds for the [[Crow-AMSAA (NHPP)\|Crow-AMSAA (NHPP)]] 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-AMSAA (NHPP)\|Crow-AMSAA (NHPP)]] 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.

	''Note regarding the Crow Bounds calculations: The equations that involve the use of the ~~Chi~~-~~Squared~~ distribution assume left-tail probability.''		''Note regarding the Crow Bounds calculations: The equations that involve the use of the chi-squared distribution assume left-tail probability.''

	==Individual (Non-Grouped) Data==		==Individual (Non-Grouped) Data==
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	:<math>C{{B}_{\beta }}=\hat{\beta }{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{\beta })}/\hat{\beta }}}\,\!</math>		:<math>C{{B}_{\beta }}=\hat{\beta }{{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\hat{\beta })}/\hat{\beta }}}\,\!</math>

	<math>\alpha \,\!</math> in <math>{{z}_{\alpha }}\,\!</math> is different ( <math>\alpha /2\,\!</math>, <math>\alpha \,\!</math> ) according to a 2-sided confidence interval or a 1-sided confidence interval, and variances can be calculated using the Fisher ~~Matrix~~.		<math>\alpha \,\!</math> in <math>{{z}_{\alpha }}\,\!</math> is different ( <math>\alpha /2\,\!</math>, <math>\alpha \,\!</math> ) according to a 2-sided confidence interval or a 1-sided confidence interval, and variances can be calculated using the Fisher matrix.

	:<math>\left[ \begin{matrix}		:<math>\left[ \begin{matrix}
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	===Growth Rate===		===Growth Rate===
	Since the growth rate, <math>\alpha \,\!</math>, is equal to <math>1-\beta \,\!</math>, the confidence bounds for both the Fisher ~~Matrix~~ and Crow methods are:		Since the growth rate, <math>\alpha \,\!</math>, is equal to <math>1-\beta \,\!</math>, the confidence bounds for both the Fisher matrix and Crow methods are:
	<br>		<br>

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

	where <math>IFI{{(t)}_{L}}\,\!</math> and <math>IFI{{(t)}_{U}}\,\!</math> are calculated using the process for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_7\|instantaneous failure intensity]].		where <math>IFI{{(t)}_{L}}\,\!</math> and <math>IFI{{(t)}_{U}}\,\!</math> are calculated using the process for calculating the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_7\|instantaneous failure intensity]].

	===Cumulative Failure Intensity===		===Cumulative Failure Intensity===
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	where <math>CFI_L\,\!</math> and <math>CFI_U\,\!</math> are calculated using the process for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_4\|cumulative failure intensity]].		where <math>CFI_L\,\!</math> and <math>CFI_U\,\!</math> are calculated using the process for calculating the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_4\|cumulative failure intensity]].

	===Instantaneous MTBF===		===Instantaneous MTBF===
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	:<math>\hat{t}={{\left( \frac{CFI}{{\hat{\lambda }}} \right)}^{\tfrac{1}{\hat{\beta }-1}}}\,\!</math>		:<math>\hat{t}={{\left( \frac{CFI}{{\hat{\lambda }}} \right)}^{\tfrac{1}{\hat{\beta }-1}}}\,\!</math>

	Then estimate, the number of failures, <math>N\,\!</math>, such that:		Then estimate the number of failures, <math>N\,\!</math>, such that:

	:<math>N=\hat{\lambda }{{\hat{t}}^{{\hat{\beta }}}}\,\!</math>		:<math>N=\hat{\lambda }{{\hat{t}}^{{\hat{\beta }}}}\,\!</math>
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	====Crow Bounds====		====Crow Bounds====
	The 2-sided confidence bounds on time given cumulative MTBF <math>(CMTBF)\,\!</math> are estimated using the process for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_8\|time given cumulative failure intensity]] <math>(CFI)\,\!</math> where <math>CFI=\frac{1}{CMTBF}\,\!</math>.		The 2-sided confidence bounds on time given cumulative MTBF <math>(CMTBF)\,\!</math> are estimated using the process for calculating the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_8\|time given cumulative failure intensity]] <math>(CFI)\,\!</math> where <math>CFI=\frac{1}{CMTBF}\,\!</math>.

	===Time Given Instantaneous MTBF===		===Time Given Instantaneous MTBF===
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	====Crow Bounds====		====Crow Bounds====
	The 2-sided confidence bounds on time given instantaneous failure intensity <math>(IFI)\,\!</math> are estimated using the process for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_10\|time given instantaneous MTBF]] where <math>IMTBF=\frac{1}{IFI}\,\!</math>.		The 2-sided confidence bounds on time given instantaneous failure intensity <math>(IFI)\,\!</math> are estimated using the process for calculating the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_10\|time given instantaneous MTBF]] where <math>IMTBF=\frac{1}{IFI}\,\!</math>.

	==Grouped Data==		==Grouped Data==
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	:<math>\hat{\beta }\,\!</math> can be obtained by <math>\underset{i=1}{\overset{K}{\mathop{\sum }}}\,{{n}_{i}}\left( \tfrac{T_{i}^{{\hat{\beta }}}\ln {{T}_{i}}-T_{i-1}^{{\hat{\beta }}}\ln \,{{T}_{i-1}}}{T_{i}^{{\hat{\beta }}}-T_{i-1}^{{\hat{\beta }}}}-\ln {{T}_{k}} \right)=0\,\!</math>.		:<math>\hat{\beta }\,\!</math> can be obtained by <math>\underset{i=1}{\overset{K}{\mathop{\sum }}}\,{{n}_{i}}\left( \tfrac{T_{i}^{{\hat{\beta }}}\ln {{T}_{i}}-T_{i-1}^{{\hat{\beta }}}\ln \,{{T}_{i-1}}}{T_{i}^{{\hat{\beta }}}-T_{i-1}^{{\hat{\beta }}}}-\ln {{T}_{k}} \right)=0\,\!</math>.

	All variance can be calculated using the Fisher ~~Matrix~~:		All variance can be calculated using the Fisher matrix:

	:<math>\left[ \begin{matrix}		:<math>\left[ \begin{matrix}
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	===Growth Rate (Grouped)===		===Growth Rate (Grouped)===
	Since the growth rate, <math>\alpha \,\!</math>, is equal to <math>1-\beta \,\!</math>, the confidence bounds for both the Fisher ~~Matrix~~ and Crow methods are:		Since the growth rate, <math>\alpha \,\!</math>, is equal to <math>1-\beta \,\!</math>, the confidence bounds for both the Fisher matrix and Crow methods are:
	<br>		<br>

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	:<math>CMTB{{F}_{U}}=\frac{1}{CF{{I}_{L}}}\,\!</math>		:<math>CMTB{{F}_{U}}=\frac{1}{CF{{I}_{L}}}\,\!</math>

	where <math>CFI_{L}\,\!</math> and <math>CFI_{U}\,\!</math> are calculating using the process for the confidence bounds on the [[Crow-AMSAA_Confidence_Bounds#Cumulative_Failure_Intensity_.28Grouped.29\|cumulative failure intensity]].		where <math>CFI_{L}\,\!</math> and <math>CFI_{U}\,\!</math> are calculating using the process for calculating the confidence bounds on the [[Crow-AMSAA_Confidence_Bounds#Cumulative_Failure_Intensity_.28Grouped.29\|cumulative failure intensity]].

	===Instantaneous MTBF (Grouped)===		===Instantaneous MTBF (Grouped)===
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	where <math>IMTB{{F}_{L}}\,\!</math>and <math>IMTB{{F}_{U}}\,\!</math> are calculated using the process for the confidence bounds on the [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_17\|instantaneous MTBF]].		where <math>IMTB{{F}_{L}}\,\!</math>and <math>IMTB{{F}_{U}}\,\!</math> are calculated using the process for calculating the confidence bounds on the [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_17\|instantaneous MTBF]].

	===Time Given Cumulative Failure Intensity (Grouped)===		===Time Given Cumulative Failure Intensity (Grouped)===
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	====Crow Bounds====		====Crow Bounds====
	The 2-sided confidence bounds on time given cumulative MTBF <math>(CMTBF)\,\!</math> are estimated using the process for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_19\|time given cumulative failure intensity]] <math>(CFI)\,\!</math> where <math>CFI=\frac{1}{CMTBF}\,\!</math>.		The 2-sided confidence bounds on time given cumulative MTBF <math>(CMTBF)\,\!</math> are estimated using the process for calculating the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_19\|time given cumulative failure intensity]] <math>(CFI)\,\!</math> where <math>CFI=\frac{1}{CMTBF}\,\!</math>.

	===Time Given Instantaneous MTBF (Grouped)===<!-- THIS SECTION HEADER IS LINKED FROM ANOTHER SECTION IN THIS PAGE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->		===Time Given Instantaneous MTBF (Grouped)===<!-- THIS SECTION HEADER IS LINKED FROM ANOTHER SECTION IN THIS PAGE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->
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	====Crow Bounds====		====Crow Bounds====
	The 2-sided confidence bounds on time given instantaneous failure intensity <math>(IFI)\,\!</math> are estimated using the process for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_21\|time given instantaneous MTBF]] where <math>IMTBF=\frac{1}{IFI}\,\!</math>.		The 2-sided confidence bounds on time given instantaneous failure intensity <math>(IFI)\,\!</math> are estimated using the process for calculating the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_21\|time given instantaneous MTBF]] where <math>IMTBF=\frac{1}{IFI}\,\!</math>.

David J. Groebel: /* Crow Bounds */

2014-04-21T18:49:09Z

Crow Bounds

← Older revision		Revision as of 18:49, 21 April 2014
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	====Crow Bounds====		====Crow Bounds====
	~~Step 1: Calculate the confidence bounds on the instantaneous MTBF:~~		'''Failure Terminated'''

	:<math>~~MTB{{F}_{i}}={{~~\~~hat{m}}_{i}}(1~~\~~pm W)~~\,\!</math>		Calculate the constants <math>p_1\,\!</math> and <math>p_2\,\!</math> using procedures described for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_17\|instantaneous MTBF]]. The lower and upper confidence bounds on time are then given by:

	~~Step~~ 2~~: Use equations in~~ the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#~~Time_Given_Instantaneous_MTBF_.28Grouped.29~~\|~~Time Given Instantaneous~~ MTBF]] ~~to calculate the~~ time given ~~the instantaneous MTBF.~~		:<math>{{\hat{t}}_{L}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{p}_{1}}} \right)}^{\tfrac{1}{1-\beta }}}</math>

			:<math>{{\hat{t}}_{U}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{p}_{2}}} \right)}^{\tfrac{1}{1-\beta }}}</math>

			'''Time Terminated'''

			Calculate the constants <math>{{\Pi }_{1}}\,\!</math> and <math>{{\Pi }_{2}}\,\!</math> using procedures described for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_17\|instantaneous MTBF]]. The lower and upper confidence bounds on time are then given by:

			:<math>{{\hat{t}}_{L}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{\Pi }_{1}}} \right)}^{\tfrac{1}{1-\beta }}}\,\!</math>

			:<math>{{\hat{t}}_{U}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{\Pi }_{2}}} \right)}^{\tfrac{1}{1-\beta }}}\,\!</math>

	===Time Given Instantaneous Failure Intensity (Grouped)===		===Time Given Instantaneous Failure Intensity (Grouped)===

David J. Groebel: /* Crow Bounds */

2014-04-18T19:18:36Z

Crow Bounds

← Older revision		Revision as of 19:18, 18 April 2014
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	If the unbiased value <math>\bar{\beta }\,\!</math> is used then:		If the unbiased value <math>\bar{\beta }\,\!</math> is used then:

	:<math>IMTBF=IMTBF\cdot \frac{N-2}{N}\,\!</math>		:<math>IMTBF=IMTBF\cdot \frac{N-1}{N}\,\!</math>

	where:		where:

David J. Groebel: /* Crow Bounds */

2014-04-18T19:18:11Z

Crow Bounds

← Older revision		Revision as of 19:18, 18 April 2014
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	If the unbiased value <math>\bar{\beta }\,\!</math> is used then:		If the unbiased value <math>\bar{\beta }\,\!</math> is used then:

	:<math>IMTBF=IMTBF\cdot \frac{N-1}{N}\,\!</math>		:<math>IMTBF=IMTBF\cdot \frac{N-2}{N}\,\!</math>

	where:		where:

David J. Groebel: /* Crow Bounds */

2014-04-16T00:04:25Z

Crow Bounds

← Older revision		Revision as of 00:04, 16 April 2014
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	====Crow Bounds====		====Crow Bounds====
	The 2-sided confidence bounds on time given instantaneous failure intensity <math>(IFI)\,\!</math> are estimated using the process for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#~~Time_Given_Instantaneous_MTBF_.28Grouped.29~~\|time given instantaneous MTBF]] where <math>IMTBF=\frac{1}{IFI}\,\!</math>.		The 2-sided confidence bounds on time given instantaneous failure intensity <math>(IFI)\,\!</math> are estimated using the process for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_21\|time given instantaneous MTBF]] where <math>IMTBF=\frac{1}{IFI}\,\!</math>.

David J. Groebel: /* Crow Bounds */

2014-04-16T00:03:29Z

Crow Bounds

← Older revision		Revision as of 00:03, 16 April 2014
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	where <math>IMTB{{F}_{L}}\,\!</math>and <math>IMTB{{F}_{U}}\,\!</math> are calculated using the process for the confidence bounds on the [[Crow-AMSAA_Confidence_Bounds#~~Instantaneous_MTBF_.28Grouped.29~~\|instantaneous MTBF]].		where <math>IMTB{{F}_{L}}\,\!</math>and <math>IMTB{{F}_{U}}\,\!</math> are calculated using the process for the confidence bounds on the [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_17\|instantaneous MTBF]].

	===Time Given Cumulative Failure Intensity (Grouped)===		===Time Given Cumulative Failure Intensity (Grouped)===

David J. Groebel: /* Crow Bounds */

2014-04-16T00:02:34Z

Crow Bounds

← Older revision		Revision as of 00:02, 16 April 2014
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	====Crow Bounds====		====Crow Bounds====
	The 2-sided confidence bounds on time given cumulative MTBF <math>(CMTBF)\,\!</math> are estimated using the process for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#~~Time_Given_Cumulative_Failure_Intensity_.28Grouped.29~~\|time given cumulative failure intensity]] <math>(CFI)\,\!</math> where <math>CFI=\frac{1}{CMTBF}\,\!</math>.		The 2-sided confidence bounds on time given cumulative MTBF <math>(CMTBF)\,\!</math> are estimated using the process for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_19\|time given cumulative failure intensity]] <math>(CFI)\,\!</math> where <math>CFI=\frac{1}{CMTBF}\,\!</math>.

	===Time Given Instantaneous MTBF (Grouped)===<!-- THIS SECTION HEADER IS LINKED FROM ANOTHER SECTION IN THIS PAGE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->		===Time Given Instantaneous MTBF (Grouped)===<!-- THIS SECTION HEADER IS LINKED FROM ANOTHER SECTION IN THIS PAGE. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK(S). -->

David J. Groebel: /* Crow Bounds */

2014-04-16T00:00:27Z

Crow Bounds

← Older revision		Revision as of 00:00, 16 April 2014
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	:<math>N{{(t)}_{U}}=\frac{t}{{\hat{\beta }}}IF{{I}_{U}}\,\!</math>		:<math>N{{(t)}_{U}}=\frac{t}{{\hat{\beta }}}IF{{I}_{U}}\,\!</math>

	where <math>IFI_L\,\!</math> and <math>IFI_U\,\!</math> are calculated based on the procedures for the confidence bounds on the [[Crow-AMSAA_Confidence_Bounds#~~Instantaneous_Failure_Intensity_.28Grouped.29~~\|instantaneous failure intensity]].		where <math>IFI_L\,\!</math> and <math>IFI_U\,\!</math> are calculated based on the procedures for the confidence bounds on the [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_18\|instantaneous failure intensity]].

	===Cumulative Failure Intensity (Grouped)===		===Cumulative Failure Intensity (Grouped)===

David J. Groebel: /* Crow Bounds */

2014-04-15T23:58:14Z

Crow Bounds

← Older revision		Revision as of 23:58, 15 April 2014
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	====Crow Bounds====		====Crow Bounds====
	The 2-sided confidence bounds on time given instantaneous failure intensity <math>(IFI)\,\!</math> are estimated using the process for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#~~Time_Given_Instantaneous_MTBF~~\|time given instantaneous MTBF]] where <math>IMTBF=\frac{1}{IFI}\,\!</math>.		The 2-sided confidence bounds on time given instantaneous failure intensity <math>(IFI)\,\!</math> are estimated using the process for the confidence bounds on [[Crow-AMSAA_Confidence_Bounds#Crow_Bounds_10\|time given instantaneous MTBF]] where <math>IMTBF=\frac{1}{IFI}\,\!</math>.

	==Grouped Data==		==Grouped Data==