Template:Weibull parameters maximum likelihood estimation - Revision history

Lisa Hacker: Redirected page to The Weibull Distribution

2012-08-03T09:08:04Z

Redirected page to The Weibull Distribution

← Older revision		Revision as of 09:08, 3 August 2012
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	~~=== Maximum Likelihood Estimation ===~~		#REDIRECT [[The Weibull Distribution]]

	~~As outlined in Chapter~~ [[Parameter Estimation]], maximum likelihood estimation works by developing a likelihood function based on the available data and finding the values of the parameter estimates that maximize the likelihood function. This can be achieved by using iterative methods to determine the parameter estimate values that maximize the likelihood function, but this can be rather difficult and time-consuming, particularly when dealing with the three-parameter distribution. Another method of finding the parameter estimates involves taking the partial derivatives of the likelihood function with respect to the parameters, setting the resulting equations equal to zero and solving simultaneously to determine the values of the parameter estimates. ( Note that MLE asymptotic properties do not hold when estimating <span class="texhtml">γ</span> using MLE [[Appendix: Weibull References\|[27]]].) The ~~log-likelihood functions and associated partial derivatives used to determine maximum likelihood estimates for the~~ Weibull ~~distribution are covered in [[Appendix:~~ Distribution ~~Log-Likelihood Equations\|Appendix~~]].


	~~'''Example 5:'''~~
	~~{{Example: Weibull MLE}}~~

Harry Guo: /* Maximum Likelihood Estimation */

2012-02-10T22:58:27Z

Maximum Likelihood Estimation

← Older revision		Revision as of 22:58, 10 February 2012
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	=== Maximum Likelihood Estimation ===		=== Maximum Likelihood Estimation ===

	As outlined in [[Parameter Estimation~~\|Chapter 4~~]], maximum likelihood estimation works by developing a likelihood function based on the available data and finding the values of the parameter estimates that maximize the likelihood function. This can be achieved by using iterative methods to determine the parameter estimate values that maximize the likelihood function, but this can be rather difficult and time-consuming, particularly when dealing with the three-parameter distribution. Another method of finding the parameter estimates involves taking the partial derivatives of the likelihood function with respect to the parameters, setting the resulting equations equal to zero and solving simultaneously to determine the values of the parameter estimates. ( Note that MLE asymptotic properties do not hold when estimating <span class="texhtml">γ</span> using MLE [[Appendix: Weibull References\|[27]]].) The log-likelihood functions and associated partial derivatives used to determine maximum likelihood estimates for the Weibull distribution are covered in [[Appendix: Distribution Log-Likelihood Equations\|Appendix]].		As outlined in Chapter [[Parameter Estimation]], maximum likelihood estimation works by developing a likelihood function based on the available data and finding the values of the parameter estimates that maximize the likelihood function. This can be achieved by using iterative methods to determine the parameter estimate values that maximize the likelihood function, but this can be rather difficult and time-consuming, particularly when dealing with the three-parameter distribution. Another method of finding the parameter estimates involves taking the partial derivatives of the likelihood function with respect to the parameters, setting the resulting equations equal to zero and solving simultaneously to determine the values of the parameter estimates. ( Note that MLE asymptotic properties do not hold when estimating <span class="texhtml">γ</span> using MLE [[Appendix: Weibull References\|[27]]].) The log-likelihood functions and associated partial derivatives used to determine maximum likelihood estimates for the Weibull distribution are covered in [[Appendix: Distribution Log-Likelihood Equations\|Appendix]].


	'''Example 5:'''		'''Example 5:'''
	{{Example: Weibull MLE}}		{{Example: Weibull MLE}}

Harry Guo at 22:44, 9 February 2012

2012-02-09T22:44:38Z

← Older revision		Revision as of 22:44, 9 February 2012
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	=== Maximum Likelihood Estimation ===		=== Maximum Likelihood Estimation ===

	As outlined in [[Parameter Estimation\|Chapter 3]], maximum likelihood estimation works by developing a likelihood function based on the available data and finding the values of the parameter estimates that maximize the likelihood function. This can be achieved by using iterative methods to determine the parameter estimate values that maximize the likelihood function, but this can be rather difficult and time-consuming, particularly when dealing with the three-parameter distribution. Another method of finding the parameter estimates involves taking the partial derivatives of the likelihood function with respect to the parameters, setting the resulting equations equal to zero and solving simultaneously to determine the values of the parameter estimates. ( Note that MLE asymptotic properties do not hold when estimating <span class="texhtml">γ</span> using MLE [27].) The log-likelihood functions and associated partial derivatives used to determine maximum likelihood estimates for the Weibull distribution are covered in Appendix C.		As outlined in [[Parameter Estimation\|Chapter 4]], maximum likelihood estimation works by developing a likelihood function based on the available data and finding the values of the parameter estimates that maximize the likelihood function. This can be achieved by using iterative methods to determine the parameter estimate values that maximize the likelihood function, but this can be rather difficult and time-consuming, particularly when dealing with the three-parameter distribution. Another method of finding the parameter estimates involves taking the partial derivatives of the likelihood function with respect to the parameters, setting the resulting equations equal to zero and solving simultaneously to determine the values of the parameter estimates. ( Note that MLE asymptotic properties do not hold when estimating <span class="texhtml">γ</span> using MLE [[Appendix: Weibull References\|[27]]].) The log-likelihood functions and associated partial derivatives used to determine maximum likelihood estimates for the Weibull distribution are covered in [[Appendix: Distribution Log-Likelihood Equations\|Appendix]].


	'''Example 5:'''		'''Example 5:'''
	{{Example: Weibull MLE}}		{{Example: Weibull MLE}}

Harry Guo: /* Maximum Likelihood Estimation */

2012-02-09T22:42:56Z

Maximum Likelihood Estimation

← Older revision		Revision as of 22:42, 9 February 2012
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	=== Maximum Likelihood Estimation ===		=== Maximum Likelihood Estimation ===

	As outlined in Chapter 3, maximum likelihood estimation works by developing a likelihood function based on the available data and finding the values of the parameter estimates that maximize the likelihood function. This can be achieved by using iterative methods to determine the parameter estimate values that maximize the likelihood function, but this can be rather difficult and time-consuming, particularly when dealing with the three-parameter distribution. Another method of finding the parameter estimates involves taking the partial derivatives of the likelihood function with respect to the parameters, setting the resulting equations equal to zero and solving simultaneously to determine the values of the parameter estimates. ( Note that MLE asymptotic properties do not hold when estimating <span class="texhtml">γ</span> using MLE [27].) The log-likelihood functions and associated partial derivatives used to determine maximum likelihood estimates for the Weibull distribution are covered in Appendix C.		As outlined in [[Parameter Estimation\|Chapter 3]], maximum likelihood estimation works by developing a likelihood function based on the available data and finding the values of the parameter estimates that maximize the likelihood function. This can be achieved by using iterative methods to determine the parameter estimate values that maximize the likelihood function, but this can be rather difficult and time-consuming, particularly when dealing with the three-parameter distribution. Another method of finding the parameter estimates involves taking the partial derivatives of the likelihood function with respect to the parameters, setting the resulting equations equal to zero and solving simultaneously to determine the values of the parameter estimates. ( Note that MLE asymptotic properties do not hold when estimating <span class="texhtml">γ</span> using MLE [27].) The log-likelihood functions and associated partial derivatives used to determine maximum likelihood estimates for the Weibull distribution are covered in Appendix C.


	'''Example 5:'''		'''Example 5:'''
	{{Example: Weibull MLE}}		{{Example: Weibull MLE}}

Harry Guo: /* Maximum Likelihood Estimation */

2012-02-09T21:01:56Z

Maximum Likelihood Estimation

← Older revision		Revision as of 21:01, 9 February 2012
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	'''Example 5:'''		'''Example 5:'''

	{{Example: Weibull MLE}}		{{Example: Weibull MLE}}

Harry Guo: /* Maximum Likelihood Estimation */

2012-02-09T21:01:14Z

Maximum Likelihood Estimation

← Older revision		Revision as of 21:01, 9 February 2012
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	{{Example: Weibull MLE}}		{{Example: Weibull MLE}}

	~~'''Maximum Likelihood Estimation Example'''~~
	~~Repeat 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 C 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>~~

	~~<br>~~
	~~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:weibullfolio16plot.png\|thumb\|center\|400px\| ]]~~

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

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


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


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

Harry Guo at 21:00, 9 February 2012

2012-02-09T21:00:54Z

← Older revision		Revision as of 21:00, 9 February 2012
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	~~====~~Example 5: Maximum Likelihood Estimation Example~~====~~		'''Example 5:'''

			{{Example: Weibull MLE}}

			'''Maximum Likelihood Estimation Example'''
	Repeat Example 1 using maximum likelihood estimation.		Repeat Example 1 using maximum likelihood estimation.

Harry Guo: Undo revision 14529 by Harry Guo (Talk)

2012-02-09T20:56:48Z

Undo revision 14529 by Harry Guo (Talk)

@@ Line 3: / Line 3: @@
 As outlined in Chapter 3, maximum likelihood estimation works by developing a likelihood function based on the available data and finding the values of the parameter estimates that maximize the likelihood function.  This can be achieved by using iterative methods to determine the parameter estimate values that maximize the likelihood function, but this can be rather difficult and time-consuming, particularly when dealing with the three-parameter distribution.  Another method of finding the parameter estimates involves taking the partial derivatives of the likelihood function with respect to the parameters, setting the resulting equations equal to zero and solving simultaneously to determine the values of the parameter estimates. ( Note that MLE asymptotic properties do not hold when estimating <span class="texhtml">γ</span> using MLE [27].) The log-likelihood functions and associated partial derivatives used to determine maximum likelihood estimates for the Weibull distribution are covered in Appendix C.
-'''Example 5:'''
-{{Example: Weibull parameters maximum likelihood estimation}}
+====Example 5: Maximum Likelihood Estimation Example====

Harry Guo: moved Template:Example: Weibull parameters maximum likelihood estimation to Template:Weibull parameters maximum likelihood estimation over redirect

2012-02-09T20:50:10Z

moved Template:Example: Weibull parameters maximum likelihood estimation to Template:Weibull parameters maximum likelihood estimation over redirect

← Older revision	Revision as of 20:50, 9 February 2012
(No difference)

Harry Guo at 20:47, 9 February 2012

2012-02-09T20:47:31Z

@@ Line 3: / Line 3: @@
 As outlined in Chapter 3, maximum likelihood estimation works by developing a likelihood function based on the available data and finding the values of the parameter estimates that maximize the likelihood function.  This can be achieved by using iterative methods to determine the parameter estimate values that maximize the likelihood function, but this can be rather difficult and time-consuming, particularly when dealing with the three-parameter distribution.  Another method of finding the parameter estimates involves taking the partial derivatives of the likelihood function with respect to the parameters, setting the resulting equations equal to zero and solving simultaneously to determine the values of the parameter estimates. ( Note that MLE asymptotic properties do not hold when estimating <span class="texhtml">γ</span> using MLE [27].) The log-likelihood functions and associated partial derivatives used to determine maximum likelihood estimates for the Weibull distribution are covered in Appendix C.
+'''Example 5:'''
-====Example 5: Maximum Likelihood Estimation Example====
+{{Example: Weibull parameters maximum likelihood estimation}}