Crow-AMSAA Confidence Bounds Example - Revision history

Lisa Hacker at 21:31, 18 September 2023

2023-09-18T21:31:29Z

← Older revision		Revision as of 21:31, 18 September 2023
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	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [~~[Crow-AMSAA (NHPP)\|~~Reliability ~~Growth and Repairable System Analysis Reference]~~]''.		''This example appears in the [https://help.reliasoft.com/reference/reliability_growth_and_repairable_system_analysis Reliability growth reference]''.
	</noinclude>		</noinclude>

David J. Groebel at 18:29, 30 May 2014

2014-05-30T18:29:49Z

← Older revision		Revision as of 18:29, 30 May 2014
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	The following figures display plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous failure intensity, respectively.		The following figures display plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous failure intensity, respectively.

	[[Image:rga5.2.png\|center\|~~400px\|Cumulative failure intensity with 2-sided 90% Fisher Matrix confidence bounds.~~]]		[[Image:rga5.2.png\|center\|450px]]

	[[Image:rga5.3.png\|center\|~~400px\|Instantaneous failure intensity with 2-sided 90% Fisher Matrix confidence bounds.~~]]		[[Image:rga5.3.png\|center\|450px]]

	'''Crow Bounds'''		'''Crow Bounds'''
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	The following figures display plots of the Crow confidence bounds for the cumulative and instantaneous failure intensity, respectively.		The following figures display plots of the Crow confidence bounds for the cumulative and instantaneous failure intensity, respectively.

	[[Image:rga5.4.png\|center\|~~400px\|Cumulative failure intensity with 2-sided 90% Crow confidence bounds.~~]]		[[Image:rga5.4.png\|center\|450px]]

	[[Image:rga5.5.png\|center\|~~400px\|Instantaneous failure intensity with 2-sided 90% Crow confidence bounds.~~]]		[[Image:rga5.5.png\|center\|450px]]

	====Example - Confidence Bounds on MTBF====		====Example - Confidence Bounds on MTBF====
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	The following two figures show plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous MTBFs.		The following two figures show plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous MTBFs.

	[[Image:rga5.6.png\|center\|~~400px\|Cumulative MTBF with 2-sided 90% Fisher Matrix confidence bounds.~~]]		[[Image:rga5.6.png\|center\|450px]]

	[[Image:rga5.7.png\|center\|~~400px\|Instantaneous MTBF with 2-sided Fisher Matrix confidence bounds.~~]]		[[Image:rga5.7.png\|center\|450px]]

	'''Crow Bounds'''		'''Crow Bounds'''
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	The figures below show plots of the Crow confidence bounds for the cumulative and instantaneous MTBF.		The figures below show plots of the Crow confidence bounds for the cumulative and instantaneous MTBF.

	[[Image:rga5.8.png\|center\|~~400px\|Cumulative MTBF with 2-sided 90% Crow confidence bounds.~~]]		[[Image:rga5.8.png\|center\|450px]]

	[[Image:rga5.9.png\|center\|~~400px\|Instantaneous MTBF with 2-sided 90% Crow confidence bounds.~~]]		[[Image:rga5.9.png\|center\|450px]]

	Confidence bounds can also be obtained on the parameters <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math>. For Fisher Matrix confidence bounds:		Confidence bounds can also be obtained on the parameters <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math>. For Fisher Matrix confidence bounds:

David J. Groebel at 00:09, 23 April 2014

2014-04-23T00:09:03Z

David J. Groebel: /* Example - Confidence Bounds on MTBF */

2014-02-04T15:49:59Z

Example - Confidence Bounds on MTBF

← Older revision		Revision as of 15:49, 4 February 2014
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	[[Image:rga5.5.png\|center\|500px\|Instantaneous failure intensity with 2-sided 90% Crow confidence bounds.]]		[[Image:rga5.5.png\|center\|500px\|Instantaneous failure intensity with 2-sided 90% Crow confidence bounds.]]

	===Example - Confidence Bounds on MTBF===		====Example - Confidence Bounds on MTBF====
	Calculate the confidence bounds on the cumulative and instantaneous MTBF for the data from the example given above.		Calculate the confidence bounds on the cumulative and instantaneous MTBF for the data from the example given above.

David J. Groebel at 15:49, 4 February 2014

2014-02-04T15:49:15Z

← Older revision		Revision as of 15:49, 4 February 2014
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	</noinclude>		</noinclude>

	===Example - Confidence Bounds on Failure Intensity===		====Example - Confidence Bounds on Failure Intensity====
	Using the values of <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> estimated in the <noinclude>[[Crow-AMSAA Parameter Estimation Example]]</noinclude><includeonly>example given above</includeonly>, calculate the 90% 2-sided confidence bounds on the cumulative and instantaneous failure intensity.		Using the values of <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> estimated in the <noinclude>[[Crow-AMSAA Parameter Estimation Example]]</noinclude><includeonly>example given above</includeonly>, calculate the 90% 2-sided confidence bounds on the cumulative and instantaneous failure intensity.

David J. Groebel at 15:41, 4 February 2014

2014-02-04T15:41:30Z

← Older revision		Revision as of 15:41, 4 February 2014
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	</noinclude>		</noinclude>

	====Confidence Bounds on ~~Cumulative and Instantaneous~~ Failure Intensity====		===Example - Confidence Bounds on Failure Intensity===
	Using the values of <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> estimated in the <noinclude>[[Crow-AMSAA Parameter Estimation Example]]</noinclude><includeonly>example given above</includeonly>, calculate the 90% 2-sided confidence bounds on the cumulative and instantaneous failure intensity.		Using the values of <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> estimated in the <noinclude>[[Crow-AMSAA Parameter Estimation Example]]</noinclude><includeonly>example given above</includeonly>, calculate the 90% 2-sided confidence bounds on the cumulative and instantaneous failure intensity.

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	[[Image:rga5.5.png\|center\|500px\|Instantaneous failure intensity with 2-sided 90% Crow confidence bounds.]]		[[Image:rga5.5.png\|center\|500px\|Instantaneous failure intensity with 2-sided 90% Crow confidence bounds.]]

			===Example - Confidence Bounds on MTBF===
	====Confidence Bounds on ~~Cumulative and Instantaneous~~ MTBF====
	Calculate the confidence bounds on the cumulative and instantaneous MTBF for the data from the example given above.		Calculate the confidence bounds on the cumulative and instantaneous MTBF for the data from the example given above.

Kate Racaza at 17:29, 31 January 2014

2014-01-31T17:29:41Z

← Older revision		Revision as of 17:29, 31 January 2014
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	====Confidence Bounds on Cumulative and Instantaneous Failure Intensity====		====Confidence Bounds on Cumulative and Instantaneous Failure Intensity====
	Using the values of <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> estimated in the <noinclude>[[Crow-AMSAA Parameter Estimation Example]]</noinclude><includeonly>example given above</includeonly>, calculate the 90% 2-sided confidence bounds on the c.		Using the values of <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> estimated in the <noinclude>[[Crow-AMSAA Parameter Estimation Example]]</noinclude><includeonly>example given above</includeonly>, calculate the 90% 2-sided confidence bounds on the cumulative and instantaneous failure intensity.

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

Kate Racaza at 17:28, 31 January 2014

2014-01-31T17:28:29Z

← Older revision		Revision as of 17:28, 31 January 2014
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	====Confidence Bounds on Cumulative and Instantaneous Failure Intensity====		====Confidence Bounds on Cumulative and Instantaneous Failure Intensity====
	Using the values of <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> estimated in the <noinclude>[[Crow-AMSAA ~~(NHPP)#Failure_Times_-_Example_1\|Failure Times -~~ Example 1]]</noinclude><includeonly>example given above</includeonly>, calculate the 90% 2-sided confidence bounds on the c.		Using the values of <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> estimated in the <noinclude>[[Crow-AMSAA Parameter Estimation Example]]</noinclude><includeonly>example given above</includeonly>, calculate the 90% 2-sided confidence bounds on the c.

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

Kate Racaza at 23:38, 30 January 2014

2014-01-30T23:38:55Z

← Older revision		Revision as of 23:38, 30 January 2014
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	</noinclude>		</noinclude>

	Using the values of <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> estimated in the <noinclude>[[Crow-AMSAA (NHPP)#Failure_Times_-_Example_1\|Failure Times - Example 1]]</noinclude><includeonly>example given above</includeonly>, calculate the 90% 2-sided confidence bounds on the ~~cumulative and instantaneous failure intensity~~.		====Confidence Bounds on Cumulative and Instantaneous Failure Intensity====
			Using the values of <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> estimated in the <noinclude>[[Crow-AMSAA (NHPP)#Failure_Times_-_Example_1\|Failure Times - Example 1]]</noinclude><includeonly>example given above</includeonly>, calculate the 90% 2-sided confidence bounds on the c.

	'''Solution'''		'''Solution'''
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	[[Image:rga5.5.png\|center\|500px\|Instantaneous failure intensity with 2-sided 90% Crow confidence bounds.]]		[[Image:rga5.5.png\|center\|500px\|Instantaneous failure intensity with 2-sided 90% Crow confidence bounds.]]

	====~~Failure Times - Example 3~~====
			====Confidence Bounds on Cumulative and Instantaneous MTBF====
	Calculate the confidence bounds on the cumulative and instantaneous MTBF for the data from the example given above.		Calculate the confidence bounds on the cumulative and instantaneous MTBF for the data from the example given above.

David J. Groebel at 20:47, 30 January 2014

2014-01-30T20:47:06Z

← Older revision		Revision as of 20:47, 30 January 2014
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	[[Image:rga5.2.png\|center\|500px\|Cumulative failure intensity with 2-sided 90% Fisher Matrix confidence bounds.]]		[[Image:rga5.2.png\|center\|500px\|Cumulative failure intensity with 2-sided 90% Fisher Matrix confidence bounds.]]


	[[Image:rga5.3.png\|center\|500px\|Instantaneous failure intensity with 2-sided 90% Fisher Matrix confidence bounds.]]		[[Image:rga5.3.png\|center\|500px\|Instantaneous failure intensity with 2-sided 90% Fisher Matrix confidence bounds.]]
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	[[Image:rga5.4.png\|center\|500px\|Cumulative failure intensity with 2-sided 90% Crow confidence bounds.]]		[[Image:rga5.4.png\|center\|500px\|Cumulative failure intensity with 2-sided 90% Crow confidence bounds.]]


	[[Image:rga5.5.png\|center\|500px\|Instantaneous failure intensity with 2-sided 90% Crow confidence bounds.]]		[[Image:rga5.5.png\|center\|500px\|Instantaneous failure intensity with 2-sided 90% Crow confidence bounds.]]
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	[[Image:rga5.6.png\|center\|500px\|Cumulative MTBF with 2-sided 90% Fisher Matrix confidence bounds.]]		[[Image:rga5.6.png\|center\|500px\|Cumulative MTBF with 2-sided 90% Fisher Matrix confidence bounds.]]


	[[Image:rga5.7.png\|center\|500px\|Instantaneous MTBF with 2-sided Fisher Matrix confidence bounds.]]		[[Image:rga5.7.png\|center\|500px\|Instantaneous MTBF with 2-sided Fisher Matrix confidence bounds.]]
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	[[Image:rga5.8.png\|center\|500px\|Cumulative MTBF with 2-sided 90% Crow confidence bounds.]]		[[Image:rga5.8.png\|center\|500px\|Cumulative MTBF with 2-sided 90% Crow confidence bounds.]]


	[[Image:rga5.9.png\|center\|500px\|Instantaneous MTBF with 2-sided 90% Crow confidence bounds.]]		[[Image:rga5.9.png\|center\|500px\|Instantaneous MTBF with 2-sided 90% Crow confidence bounds.]]

← Older revision		Revision as of 20:47, 30 January 2014
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	[[Image:rga5.2.png\|center\|500px\|Cumulative failure intensity with 2-sided 90% Fisher Matrix confidence bounds.]]		[[Image:rga5.2.png\|center\|500px\|Cumulative failure intensity with 2-sided 90% Fisher Matrix confidence bounds.]]


	[[Image:rga5.3.png\|center\|500px\|Instantaneous failure intensity with 2-sided 90% Fisher Matrix confidence bounds.]]		[[Image:rga5.3.png\|center\|500px\|Instantaneous failure intensity with 2-sided 90% Fisher Matrix confidence bounds.]]
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	[[Image:rga5.4.png\|center\|500px\|Cumulative failure intensity with 2-sided 90% Crow confidence bounds.]]		[[Image:rga5.4.png\|center\|500px\|Cumulative failure intensity with 2-sided 90% Crow confidence bounds.]]


	[[Image:rga5.5.png\|center\|500px\|Instantaneous failure intensity with 2-sided 90% Crow confidence bounds.]]		[[Image:rga5.5.png\|center\|500px\|Instantaneous failure intensity with 2-sided 90% Crow confidence bounds.]]
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	[[Image:rga5.6.png\|center\|500px\|Cumulative MTBF with 2-sided 90% Fisher Matrix confidence bounds.]]		[[Image:rga5.6.png\|center\|500px\|Cumulative MTBF with 2-sided 90% Fisher Matrix confidence bounds.]]


	[[Image:rga5.7.png\|center\|500px\|Instantaneous MTBF with 2-sided Fisher Matrix confidence bounds.]]		[[Image:rga5.7.png\|center\|500px\|Instantaneous MTBF with 2-sided Fisher Matrix confidence bounds.]]
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	[[Image:rga5.8.png\|center\|500px\|Cumulative MTBF with 2-sided 90% Crow confidence bounds.]]		[[Image:rga5.8.png\|center\|500px\|Cumulative MTBF with 2-sided 90% Crow confidence bounds.]]


	[[Image:rga5.9.png\|center\|500px\|Instantaneous MTBF with 2-sided 90% Crow confidence bounds.]]		[[Image:rga5.9.png\|center\|500px\|Instantaneous MTBF with 2-sided 90% Crow confidence bounds.]]

← Older revision		Revision as of 00:09, 23 April 2014
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	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [[Crow-AMSAA (NHPP)\|Reliability Growth and Repairable System Analysis Reference ~~book~~]]''.		''This example appears in the [[Crow-AMSAA (NHPP)\|Reliability Growth and Repairable System Analysis Reference]]''.
	</noinclude>		</noinclude>

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	The following figures display plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous failure intensity, respectively.		The following figures display plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous failure intensity, respectively.

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

	[[Image:rga5.3.png\|center\|~~500px~~\|Instantaneous failure intensity with 2-sided 90% Fisher Matrix confidence bounds.]]		[[Image:rga5.3.png\|center\|400px\|Instantaneous failure intensity with 2-sided 90% Fisher Matrix confidence bounds.]]

	'''Crow Bounds'''		'''Crow Bounds'''
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	The following figures display plots of the Crow confidence bounds for the cumulative and instantaneous failure intensity, respectively.		The following figures display plots of the Crow confidence bounds for the cumulative and instantaneous failure intensity, respectively.

	[[Image:rga5.4.png\|center\|~~500px~~\|Cumulative failure intensity with 2-sided 90% Crow confidence bounds.]]		[[Image:rga5.4.png\|center\|400px\|Cumulative failure intensity with 2-sided 90% Crow confidence bounds.]]

	[[Image:rga5.5.png\|center\|~~500px~~\|Instantaneous failure intensity with 2-sided 90% Crow confidence bounds.]]		[[Image:rga5.5.png\|center\|400px\|Instantaneous failure intensity with 2-sided 90% Crow confidence bounds.]]

	====Example - Confidence Bounds on MTBF====		====Example - Confidence Bounds on MTBF====
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	The following two figures show plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous MTBFs.		The following two figures show plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous MTBFs.

	[[Image:rga5.6.png\|center\|~~500px~~\|Cumulative MTBF with 2-sided 90% Fisher Matrix confidence bounds.]]		[[Image:rga5.6.png\|center\|400px\|Cumulative MTBF with 2-sided 90% Fisher Matrix confidence bounds.]]

	[[Image:rga5.7.png\|center\|~~500px~~\|Instantaneous MTBF with 2-sided Fisher Matrix confidence bounds.]]		[[Image:rga5.7.png\|center\|400px\|Instantaneous MTBF with 2-sided Fisher Matrix confidence bounds.]]

	'''Crow Bounds'''		'''Crow Bounds'''
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	The figures below show plots of the Crow confidence bounds for the cumulative and instantaneous MTBF.		The figures below show plots of the Crow confidence bounds for the cumulative and instantaneous MTBF.

	[[Image:rga5.8.png\|center\|~~500px~~\|Cumulative MTBF with 2-sided 90% Crow confidence bounds.]]		[[Image:rga5.8.png\|center\|400px\|Cumulative MTBF with 2-sided 90% Crow confidence bounds.]]

	[[Image:rga5.9.png\|center\|~~500px~~\|Instantaneous MTBF with 2-sided 90% Crow confidence bounds.]]		[[Image:rga5.9.png\|center\|400px\|Instantaneous MTBF with 2-sided 90% Crow confidence bounds.]]

	Confidence bounds can also be obtained on the parameters <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math>. For Fisher Matrix confidence bounds:		Confidence bounds can also be obtained on the parameters <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math>. For Fisher Matrix confidence bounds: