Template:Example: Weibull MLE - Revision history

Lisa Hacker: Redirected page to The Weibull Distribution

2012-08-08T09:04:05Z

Redirected page to The Weibull Distribution

← Older revision		Revision as of 09:04, 8 August 2012
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	~~'''Maximum Likelihood Estimation Example'''~~		#REDIRECT [[The Weibull Distribution]]

	~~Repeat~~ [[~~Weibull Example 1 Data\|Example 1]] using maximum likelihood estimation.~~

	~~'''Solution'''~~

	In this case, we have non-grouped data with no suspensions or intervals, (i.e., complete data). The equations for the partial derivatives of the log-likelihood function are derived in [[Appendix:_Log-Likelihood_Equations\|an appendix]] and given next:
	::<math> \frac{\partial \Lambda }{\partial \beta }=\frac{6}{\beta } +\sum_{i=1}^{6}\ln \left( \frac{T_{i}}{\eta }\right) -\sum_{i=1}^{6}\left( \frac{T_{i}}{\eta }\right) ^{\beta }\ln \left( \frac{T_{i}}{\eta }\right) =0
	~~</math>~~

	~~:and:~~

	~~::<math> \frac{\partial \Lambda }{\partial \eta }=\frac{-\beta }{\eta }\cdot 6+\frac{ \beta }{\eta }\sum\limits_{i=1}^{6}\left( \frac{T_{i}}{\eta }\right) ^{\beta }=0 </math>~~

	~~Solving the above equations simultaneously we get:~~

	~~::<math> \hat{\beta }=1.933,</math> <math>\hat{\eta }=73.526 </math>~~

	~~The variance/covariance matrix is found to be,~~

	~~::<math> \left[ \begin{array}{ccc} \hat{Var}\left( \hat{\beta }\right) =0.4211 & \hat{Cov}( \hat{\beta },\hat{\eta })=3.272 \\~~

	~~\hat{Cov}(\hat{\beta },\hat{\eta })=3.272 & \hat{Var} \left( \hat{\eta }\right) =266.646 \end{array} \right] </math>~~

	The ~~results and the associated graph using Weibull++ (MLE) are shown next.~~

	~~[[Image:Weibull Distribution Example 5 Plot.png\|center\|450px\| ]]~~

	~~You can view the variance/covariance matrix directly by clicking the ''Analysis Summary'' at the '''Main''' Panel.~~

	~~[[Image:~~ Weibull Distribution ~~Example 5 Variance Matrix.png\|center\|450px\|~~ ]]

	~~Note that the decimal accuracy displayed and used is based on your individual Application Setup.~~

Richard House at 02:12, 8 August 2012

2012-08-08T02:12:38Z

← Older revision		Revision as of 02:12, 8 August 2012
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	Repeat [[Weibull Example 1 Data\|Example 1]] using maximum likelihood estimation.		Repeat [[Weibull Example 1 Data\|Example 1]] using maximum likelihood estimation.


	'''Solution'''		'''Solution'''
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	::<math> \hat{\beta }=1.933,</math> <math>\hat{\eta }=73.526 </math>		::<math> \hat{\beta }=1.933,</math> <math>\hat{\eta }=73.526 </math>

	~~<br>~~
	The variance/covariance matrix is found to be,		The variance/covariance matrix is found to be,

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	[[Image: Weibull Distribution Example 5 Variance Matrix.png\|center\|450px\| ]]		[[Image: Weibull Distribution Example 5 Variance Matrix.png\|center\|450px\| ]]

	~~<br>~~ Note that the decimal accuracy displayed and used is based on your individual ~~User~~ Setup.		Note that the decimal accuracy displayed and used is based on your individual Application Setup.

Richard House at 05:39, 6 August 2012

2012-08-06T05:39:29Z

← Older revision		Revision as of 05:39, 6 August 2012
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	The results and the associated graph using Weibull++ (MLE) are shown next.		The results and the associated graph using Weibull++ (MLE) are shown next.

	[[Image:Weibull Distribution Example 5 Plot.png~~\|thumb~~\|center\|450px\| ]]		[[Image:Weibull Distribution Example 5 Plot.png\|center\|450px\| ]]

	You can view the variance/covariance matrix directly by clicking the ''Analysis Summary'' at the '''Main''' Panel.		You can view the variance/covariance matrix directly by clicking the ''Analysis Summary'' at the '''Main''' Panel.

	[[Image: Weibull Distribution Example 5 Variance Matrix.png~~\|thumb~~\|center\|450px\| ]]		[[Image: Weibull Distribution Example 5 Variance Matrix.png\|center\|450px\| ]]

	<br> Note that the decimal accuracy displayed and used is based on your individual User Setup.		<br> Note that the decimal accuracy displayed and used is based on your individual User Setup.

Richard House at 05:08, 6 August 2012

2012-08-06T05:08:36Z

← Older revision		Revision as of 05:08, 6 August 2012
Line 27:		Line 27:
	The results and the associated graph using Weibull++ (MLE) are shown next.		The results and the associated graph using Weibull++ (MLE) are shown next.

	[[Image:Weibull Distribution Example 5 Plot.png\|thumb\|center\|~~250px~~\| ]]		[[Image:Weibull Distribution Example 5 Plot.png\|thumb\|center\|450px\| ]]

	You can view the variance/covariance matrix directly by clicking the ''Analysis Summary'' at the '''Main''' Panel.		You can view the variance/covariance matrix directly by clicking the ''Analysis Summary'' at the '''Main''' Panel.

	[[Image: Weibull Distribution Example 5 Variance Matrix.png\|thumb\|center\|~~250px~~\| ]]		[[Image: Weibull Distribution Example 5 Variance Matrix.png\|thumb\|center\|450px\| ]]

	<br> Note that the decimal accuracy displayed and used is based on your individual User Setup.		<br> Note that the decimal accuracy displayed and used is based on your individual User Setup.

Richard House at 04:51, 6 August 2012

2012-08-06T04:51:27Z

← Older revision		Revision as of 04:51, 6 August 2012
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	'''Solution'''		'''Solution'''

	In this case, we have non-grouped data with no suspensions or intervals, i.e., complete data. The equations for the partial derivatives of the log-likelihood function are derived in [[Appendix:_Log-Likelihood_Equations\|an appendix]] and given next:		In this case, we have non-grouped data with no suspensions or intervals, (i.e., complete data). The equations for the partial derivatives of the log-likelihood function are derived in [[Appendix:_Log-Likelihood_Equations\|an appendix]] and given next:
	::<math> \frac{\partial \Lambda }{\partial \beta }=\frac{6}{\beta } +\sum_{i=1}^{6}\ln \left( \frac{T_{i}}{\eta }\right) -\sum_{i=1}^{6}\left( \frac{T_{i}}{\eta }\right) ^{\beta }\ln \left( \frac{T_{i}}{\eta }\right) =0		::<math> \frac{\partial \Lambda }{\partial \beta }=\frac{6}{\beta } +\sum_{i=1}^{6}\ln \left( \frac{T_{i}}{\eta }\right) -\sum_{i=1}^{6}\left( \frac{T_{i}}{\eta }\right) ^{\beta }\ln \left( \frac{T_{i}}{\eta }\right) =0
	</math>		</math>

Lisa Hacker at 08:41, 3 August 2012

2012-08-03T08:41:42Z

← Older revision		Revision as of 08:41, 3 August 2012
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	'''Solution'''		'''Solution'''

	In this case, we have non-grouped data with no suspensions or intervals, i.e. complete data. The equations for the partial derivatives of the log-likelihood function are derived in [[Appendix: ~~Distribution Log~~-~~Likelihood Equations~~\|~~Appendix~~]] and given next:		In this case, we have non-grouped data with no suspensions or intervals, i.e., complete data. The equations for the partial derivatives of the log-likelihood function are derived in [[Appendix:_Log-Likelihood_Equations\|an appendix]] and given next:
	::<math> \frac{\partial \Lambda }{\partial \beta }=\frac{6}{\beta } +\sum_{i=1}^{6}\ln \left( \frac{T_{i}}{\eta }\right) -\sum_{i=1}^{6}\left( \frac{T_{i}}{\eta }\right) ^{\beta }\ln \left( \frac{T_{i}}{\eta }\right) =0		::<math> \frac{\partial \Lambda }{\partial \beta }=\frac{6}{\beta } +\sum_{i=1}^{6}\ln \left( \frac{T_{i}}{\eta }\right) -\sum_{i=1}^{6}\left( \frac{T_{i}}{\eta }\right) ^{\beta }\ln \left( \frac{T_{i}}{\eta }\right) =0
	</math>		</math>

Nicolette Young at 18:16, 25 April 2012

2012-04-25T18:16:22Z

← Older revision		Revision as of 18:16, 25 April 2012
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	The results and the associated graph using Weibull++ (MLE) are shown next.		The results and the associated graph using Weibull++ (MLE) are shown next.

	[[Image:Weibull Distribution Example 5 Plot.png\|thumb\|center\|~~400px~~\| ]]		[[Image:Weibull Distribution Example 5 Plot.png\|thumb\|center\|250px\| ]]

	You can view the variance/covariance matrix directly by clicking the ''Analysis Summary'' at the '''Main''' Panel.		You can view the variance/covariance matrix directly by clicking the ''Analysis Summary'' at the '''Main''' Panel.

	[[Image: Weibull Distribution Example 5 Variance Matrix.png\|thumb\|center\|~~400px~~\| ]]		[[Image: Weibull Distribution Example 5 Variance Matrix.png\|thumb\|center\|250px\| ]]

	<br> Note that the decimal accuracy displayed and used is based on your individual User Setup.		<br> Note that the decimal accuracy displayed and used is based on your individual User Setup.

Harry Guo at 23:02, 1 March 2012

2012-03-01T23:02:22Z

← Older revision		Revision as of 23:02, 1 March 2012
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	You can view the variance/covariance matrix directly by clicking the ''Analysis Summary'' at the '''Main''' Panel.		You can view the variance/covariance matrix directly by clicking the ''Analysis Summary'' at the '''Main''' Panel.

	[[Weibull Distribution Example 5 Variance Matrix.png\|thumb\|center\|400px\| ]]		[[Image: Weibull Distribution Example 5 Variance Matrix.png\|thumb\|center\|400px\| ]]

	<br> Note that the decimal accuracy displayed and used is based on your individual User Setup.		<br> Note that the decimal accuracy displayed and used is based on your individual User Setup.

Harry Guo at 23:02, 1 March 2012

2012-03-01T23:02:09Z

← Older revision		Revision as of 23:02, 1 March 2012
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	[[Image:Weibull Distribution Example 5 Plot.png\|thumb\|center\|400px\| ]]		[[Image:Weibull Distribution Example 5 Plot.png\|thumb\|center\|400px\| ]]

	You can view the variance/covariance matrix directly by clicking the ''~~Quick Calculation Pad~~ ''~~(QCP) icon~~		You can view the variance/covariance matrix directly by clicking the ''Analysis Summary'' at the '''Main''' Panel.

	~~[[Image:qcpicon.gif\|thumb\|center\|400px\| ]]~~


	~~[[Image:qcpfolio16~~.~~png\|thumb\|center\|400px\| ]]~~

			[[Weibull Distribution Example 5 Variance Matrix.png\|thumb\|center\|400px\| ]]

	<br> Note that the decimal accuracy displayed and used is based on your individual User Setup.		<br> Note that the decimal accuracy displayed and used is based on your individual User Setup.

Harry Guo at 22:59, 1 March 2012

2012-03-01T22:59:22Z

← Older revision		Revision as of 22:59, 1 March 2012
Line 27:		Line 27:
	The results and the associated graph using Weibull++ (MLE) are shown next.		The results and the associated graph using Weibull++ (MLE) are shown next.

	[[Image:~~weibullfolio16plot~~.png\|thumb\|center\|400px\| ]]		[[Image:Weibull Distribution Example 5 Plot.png\|thumb\|center\|400px\| ]]

	You can view the variance/covariance matrix directly by clicking the ''Quick Calculation Pad ''(QCP) icon		You can view the variance/covariance matrix directly by clicking the ''Quick Calculation Pad ''(QCP) icon