Weibull Distribution Examples - Revision history

Lisa Hacker at 21:47, 18 September 2023

2023-09-18T21:47:29Z

← Older revision		Revision as of 21:47, 18 September 2023
Line 1:		Line 1:
	<noinclude>{{Banner Weibull Examples}}{{Navigation box}}		<noinclude>{{Banner Weibull Examples}}{{Navigation box}}
	''These examples also appear in the [~~[The_Weibull_Distribution#Weibull Distribution Examples\|~~Life ~~Data Analysis Reference book]~~].''		''These examples also appear in the [https://help.reliasoft.com/reference/life_data_analysis Life data analysis reference].''
	</noinclude>		</noinclude>
	===Median Rank Plot Example===		===Median Rank Plot Example===

Lisa Hacker: Removed protection from "Weibull Distribution Examples"

2023-09-18T21:47:08Z

Removed protection from "Weibull Distribution Examples"

← Older revision	Revision as of 21:47, 18 September 2023
(No difference)

Richard House: /* Benchmark with Published Examples */

2012-09-25T20:37:25Z

Benchmark with Published Examples

← Older revision		Revision as of 20:37, 25 September 2012
Line 374:		Line 374:
	'''Published Results (using MLE)'''		'''Published Results (using MLE)'''

	~~[[Image~~:~~example17formulamle~~.~~png]]~~		::<math>\begin{align}
			& \widehat{\beta }=3.7596935\\
			& \widehat{\eta} = 106.49758 \\
			& \hat{\gamma }=14.451684\\
			\end{align}\,\!</math>

	'''Computed Results in Weibull++'''		'''Computed Results in Weibull++'''

	~~[[Image~~:~~example17formulamle2~~.~~png]]~~		::<math>\begin{align}
			& \widehat{\beta }=3.7596935\\
			& \widehat{\eta} = 106.49758 \\
			& \hat{\gamma }=14.451684\\
			\end{align}\,\!</math>

	Note that you must select the '''Use True 3-P MLE'''option in the Weibull++ Application Setup to replicate these results.		Note that you must select the '''Use True 3-P MLE'''option in the Weibull++ Application Setup to replicate these results.

Richard House: /* Benchmark with Published Examples */

2012-09-25T20:33:42Z

Benchmark with Published Examples

← Older revision		Revision as of 20:33, 25 September 2012
Line 327:		Line 327:
	Published results (using MLE):		Published results (using MLE):

	~~[[Image~~:~~example16formula~~.~~png]]~~		::<math>\begin{align}
			& \widehat{\beta }=1.486 \\
			& \widehat{\eta} = 71.687\\
			\end{align}\,\!</math>

	Published 95% FM confidence limits on the parameters:		Published 95% FM confidence limits on the parameters:

	~~[[Image~~:~~example16formula2~~.~~png]]~~		::<math>\begin{align}
			& \widehat{\beta }=\lbrace 1.224, \text{ }1.802\rbrace \\
			& \widehat{\eta} = \lbrace 61.962, \text{ }82.938\rbrace \\
			\end{align}\,\!</math>

	Published variance/covariance matrix:		Published variance/covariance matrix:
Line 343:		Line 349:
	Weibull++ computed parameters for maximum likelihood are:		Weibull++ computed parameters for maximum likelihood are:

	~~[[Image~~:~~compexample16formula~~.~~png]]~~		::<math>\begin{align}
			& \widehat{\beta }=1.485 \\
			& \widehat{\eta} = 71.690\\
			\end{align}\,\!</math>

	Weibull++ computed 95% FM confidence limits on the parameters:		Weibull++ computed 95% FM confidence limits on the parameters:

	~~[[Image~~:~~compexample16formula2~~.~~png]]~~		::<math>\begin{align}
			& \widehat{\beta }=\lbrace 1.224, \text{ }1.802\rbrace \\
			& \widehat{\eta} = \lbrace 61.961, \text{ }82.947\rbrace \\
			\end{align}\,\!</math>

	Weibull++ computed/variance covariance matrix:		Weibull++ computed/variance covariance matrix:

Richard House: /* Benchmark with Published Examples */

2012-09-25T20:27:29Z

Benchmark with Published Examples

← Older revision		Revision as of 20:27, 25 September 2012
Line 266:		Line 266:
	::<math>\begin{align}		::<math>\begin{align}
	& \widehat{\beta }=1.0584 \\		& \widehat{\beta }=1.0584 \\
	& \widehat{\eta} = 26.296 \\		& \widehat{\eta} = 26,296 \\
	\end{align}\,\!</math>		\end{align}\,\!</math>

Line 273:		Line 273:
	::<math>\begin{align}		::<math>\begin{align}
	& \widehat{\beta }=\lbrace 0.6441, \text{ }1.7394\rbrace \\		& \widehat{\beta }=\lbrace 0.6441, \text{ }1.7394\rbrace \\
	& \widehat{\eta} = \lbrace ~~10522~~, \text{ }65.532\rbrace \\		& \widehat{\eta} = \lbrace 10,522, \text{ }65,532\rbrace \\
	\end{align}\,\!</math>		\end{align}\,\!</math>

Line 296:		Line 296:
	Weibull++ computed parameters for maximum likelihood are:		Weibull++ computed parameters for maximum likelihood are:

	~~[[Image~~:~~compexample18formula~~.~~png]]~~		::<math>\begin{align}
			& \widehat{\beta }=1.0584 \\
			& \widehat{\eta} = 26,297 \\
			\end{align}\,\!</math>

	Weibull++ computed 95% FM confidence limits on the parameters:		Weibull++ computed 95% FM confidence limits on the parameters:

	~~[[Image~~:~~compexample18formula2~~.~~png]]~~		::<math>\begin{align}
			& \widehat{\beta }=\lbrace 0.6441, \text{ }1.7394\rbrace \\
			& \widehat{\eta} = \lbrace 10,522, \text{ }65,532\rbrace \\
			\end{align}\,\!</math>

	Weibull++ computed/variance covariance matrix:		Weibull++ computed/variance covariance matrix:

Richard House: /* Benchmark with Published Examples */

2012-09-25T20:17:49Z

Benchmark with Published Examples

← Older revision		Revision as of 20:17, 25 September 2012
Line 241:		Line 241:
	Weibull++ computed parameters for RRY are:		Weibull++ computed parameters for RRY are:

	~~[[Image~~:~~example15formula2~~.~~png]]~~		::<math>\begin{align}
			& \widehat{\beta }=1.1973 \\
			& \widehat{\eta} = 146.2545 \\
			& \hat{\rho }=0.9999\\
			\end{align}\,\!</math>

	The small difference between the published results and the ones obtained from Weibull++ is due to the difference in the median rank values between the two (in the publication, median ranks are obtained from tables to 3 decimal places, whereas in Weibull++ they are calculated and carried out up to the 15th decimal point).		The small difference between the published results and the ones obtained from Weibull++ is due to the difference in the median rank values between the two (in the publication, median ranks are obtained from tables to 3 decimal places, whereas in Weibull++ they are calculated and carried out up to the 15th decimal point).
Line 260:		Line 264:
	Weibull parameters (2P-Weibull, MLE):		Weibull parameters (2P-Weibull, MLE):

	~~[[Image~~:~~example18formula~~.~~png]]~~		::<math>\begin{align}
			& \widehat{\beta }=1.0584 \\
			& \widehat{\eta} = 26.296 \\
			\end{align}\,\!</math>

	Published 95% FM confidence limits on the parameters:		Published 95% FM confidence limits on the parameters:

	~~[[Image~~:~~example18formula2~~.~~png]]~~		::<math>\begin{align}
			& \widehat{\beta }=\lbrace 0.6441, \text{ }1.7394\rbrace \\
			& \widehat{\eta} = \lbrace 10522, \text{ }65.532\rbrace \\
			\end{align}\,\!</math>

	Published variance/covariance matrix:		Published variance/covariance matrix:

Richard House: /* Benchmark with Published Examples */

2012-09-25T20:11:50Z

Benchmark with Published Examples

← Older revision		Revision as of 20:11, 25 September 2012
Line 234:		Line 234:
	& \hat{\rho }=0.998703\\		& \hat{\rho }=0.998703\\
	\end{align}\,\!</math>		\end{align}\,\!</math>


	~~[[Image:example15formula.png]]~~

	'''Computed Results in Weibull++'''		'''Computed Results in Weibull++'''

Richard House: /* Benchmark with Published Examples */

2012-09-25T20:11:41Z

Benchmark with Published Examples

← Older revision		Revision as of 20:11, 25 September 2012
Line 228:		Line 228:

	Published Results (using Rank Regression on Y):		Published Results (using Rank Regression on Y):

			::<math>\begin{align}
			& \widehat{\beta }=1.20 \\
			& \widehat{\eta} = 146.2 \\
			& \hat{\rho }=0.998703\\
			\end{align}\,\!</math>


	[[Image:example15formula.png]]		[[Image:example15formula.png]]

Richard House: /* Mixed Data Types Example */

2012-09-25T20:08:38Z

Mixed Data Types Example

← Older revision		Revision as of 20:08, 25 September 2012
Line 197:		Line 197:
	Published results (using probability plotting):		Published results (using probability plotting):

	::~~[[Image:example19formula~~.~~png]]~~		::<math>{\widehat{\beta}} = 3.0\,\!</math>, <math>{\widehat{\eta}} = 1,220\,\!</math>, <math>{\widehat{\gamma}} = -300\,\!</math>

	'''Computed Results in Weibull++'''		'''Computed Results in Weibull++'''
Line 203:		Line 203:
	Weibull++ computed parameters for rank regression on X are:		Weibull++ computed parameters for rank regression on X are:

	::~~[[Image:compexample19formula~~.~~png]]~~		::<math>{\widehat{\beta}} = 2.9013\,\!</math>, <math>{\widehat{\eta}} = 1195.5009\,\!</math>, <math>{\widehat{\gamma}} = -279.000\,\!</math>

	The small difference between the published results and the ones obtained from Weibull++ are due to the difference in the estimation method. In the publication the parameters were estimated using probability plotting (i.e., the fitted line was "eye-balled"). In Weibull++, the parameters were estimated using non-linear regression (a more accurate, mathematically fitted line). Note that γ in this example is negative. This means that the unadjusted for γ line is concave up, as shown next.		The small difference between the published results and the ones obtained from Weibull++ are due to the difference in the estimation method. In the publication the parameters were estimated using probability plotting (i.e., the fitted line was "eye-balled"). In Weibull++, the parameters were estimated using non-linear regression (a more accurate, mathematically fitted line). Note that γ in this example is negative. This means that the unadjusted for γ line is concave up, as shown next.

Richard House at 18:32, 25 September 2012

2012-09-25T18:32:45Z

← Older revision		Revision as of 18:32, 25 September 2012
Line 12:		Line 12:
	Thus, from the F-distribution rank equation:		Thus, from the F-distribution rank equation:

	::<math>MR=\frac{1}{1+\left( \frac{10-6+1}{6} \right){{F}_{0.5;10;12}}}</math>		::<math>MR=\frac{1}{1+\left( \frac{10-6+1}{6} \right){{F}_{0.5;10;12}}}\,\!</math>

	Use the QSR to calculate the value of F<sub>0.5;10;12</sub> = 0.9886, as shown next:		Use the QSR to calculate the value of F<sub>0.5;10;12</sub> = 0.9886, as shown next:
Line 20:		Line 20:
	Consequently:		Consequently:

	::<math>MR=\frac{1}{1+\left( \frac{5}{6} \right)\times 0.9886}=0.5483=54.83%</math>		::<math>MR=\frac{1}{1+\left( \frac{5}{6} \right)\times 0.9886}=0.5483=54.83%\,\!</math>

	Another method is to use the '''Median Ranks''' option directly, which yields MR(%) = 54.8305%, as shown next:		Another method is to use the '''Median Ranks''' option directly, which yields MR(%) = 54.8305%, as shown next:
Line 27:		Line 27:
	===Complete Data Example===		===Complete Data Example===

	Assume that 10 identical units (N = 10) are being reliability tested at the same application and operation stress levels. 6 of these units fail during this test after operating the following numbers of hours, <math>{T}_{j}</math>: 150, 105, 83, 123, 64 and 46. The test is stopped at the 6th failure. Find the parameters of the Weibull ''pdf'' that represents these data.		Assume that 10 identical units (N = 10) are being reliability tested at the same application and operation stress levels. 6 of these units fail during this test after operating the following numbers of hours, <math>{T}_{j}\,\!</math>: 150, 105, 83, 123, 64 and 46. The test is stopped at the 6th failure. Find the parameters of the Weibull ''pdf'' that represents these data.

	'''Solution'''		'''Solution'''
Line 108:		Line 108:
	& \hat{\beta }=1.145 \\		& \hat{\beta }=1.145 \\
	& \hat{\eta }=65.97 \\		& \hat{\eta }=65.97 \\
	\end{align}</math>		\end{align}\,\!</math>


Line 116:		Line 116:
	& \hat{\beta }=0.914\\		& \hat{\beta }=0.914\\
	& \hat{\eta }=79.38 \\		& \hat{\eta }=79.38 \\
	\end{align}</math>		\end{align}\,\!</math>


Line 124:		Line 124:
	& \hat{\beta }=0.895\\		& \hat{\beta }=0.895\\
	& \hat{\eta }=82.02 \\		& \hat{\eta }=82.02 \\
	\end{align}</math>		\end{align}\,\!</math>

	===Interval Data Example===		===Interval Data Example===
Line 166:		Line 166:
	& \hat{\beta }=5.76 \\		& \hat{\beta }=5.76 \\
	& \hat{\eta }=44.68 \\		& \hat{\eta }=44.68 \\
	\end{align}</math>		\end{align}\,\!</math>


Line 174:		Line 174:
	& \hat{\beta }=5.70 \\		& \hat{\beta }=5.70 \\
	& \hat{\eta }=44.54 \\		& \hat{\eta }=44.54 \\
	\end{align}</math>		\end{align}\,\!</math>


Line 182:		Line 182:
	& \hat{\beta }=5.41 \\		& \hat{\beta }=5.41 \\
	& \hat{\eta }=44.76 \\		& \hat{\eta }=44.76 \\
	\end{align}</math>		\end{align}\,\!</math>

	The plot of the MLE solution with the two-sided 90% confidence bounds is:		The plot of the MLE solution with the two-sided 90% confidence bounds is:
Line 415:		Line 415:
	The computed parameters using MLE are:		The computed parameters using MLE are:

	::<math>\hat{\beta }=0.748;\text{ }\hat{\eta }=44.38</math>		::<math>\hat{\beta }=0.748;\text{ }\hat{\eta }=44.38\,\!</math>

	Using RRX:		Using RRX:

	::<math>\hat{\beta }=1.057;\text{ }\hat{\eta }=36.29</math>		::<math>\hat{\beta }=1.057;\text{ }\hat{\eta }=36.29\,\!</math>

	Using RRY:		Using RRY:

	::<math>\hat{\beta }=0.998;\text{ }\hat{\eta }=37.16</math>		::<math>\hat{\beta }=0.998;\text{ }\hat{\eta }=37.16\,\!</math>

	The plot with the two-sided 90% confidence bounds for the rank regression on X solution is:		The plot with the two-sided 90% confidence bounds for the rank regression on X solution is:
	[[Image: RRX Plot for Example 14.png\|center\|550px]]		[[Image: RRX Plot for Example 14.png\|center\|550px]]