Template:Maximum Likelihood Estimation for Exponential Distribution - Revision history

Lisa Hacker: Redirected page to The Weibull Distribution

2012-08-03T09:09:10Z

Redirected page to The Weibull Distribution

← Older revision		Revision as of 09:09, 3 August 2012
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	~~===Maximum Likelihood Estimation for Exponential Distribution===~~		#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. 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 equation with respect to the parameters, setting the resulting equations equal to zero, and solving simultaneously to determine the values of the parameter estimates. The ~~log-likelihood functions and associated partial derivatives used to determine maximum likelihood estimates for the exponential distribution are covered in [[Appendix:~~ Distribution ~~Log-Likelihood Equations\|Appendix~~]].

	~~'''Example 4:'''~~
	~~{{Example: MLE for Exponential Distribution}}~~

Harry Guo: /* Maximum Likelihood Estimation for Exponential Distribution */

2012-02-10T23:02:06Z

Maximum Likelihood Estimation for Exponential Distribution

← Older revision		Revision as of 23:02, 10 February 2012
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	===Maximum Likelihood Estimation for Exponential Distribution===		===Maximum Likelihood Estimation for Exponential Distribution===
	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. 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 equation with respect to the parameters, setting the resulting equations equal to zero, and solving simultaneously to determine the values of the parameter estimates. The log-likelihood functions and associated partial derivatives used to determine maximum likelihood estimates for the exponential 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. 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 equation with respect to the parameters, setting the resulting equations equal to zero, and solving simultaneously to determine the values of the parameter estimates. The log-likelihood functions and associated partial derivatives used to determine maximum likelihood estimates for the exponential distribution are covered in [[Appendix: Distribution Log-Likelihood Equations\|Appendix]].

	'''Example 4:'''		'''Example 4:'''
	{{Example: MLE for Exponential Distribution}}		{{Example: MLE for Exponential Distribution}}

Harry Guo: /* Maximum Likelihood Estimation for Exponential Distribution */

2012-02-09T18:24:52Z

Maximum Likelihood Estimation for Exponential Distribution

← Older revision		Revision as of 18:24, 9 February 2012
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	'''Example 4:'''		'''Example 4:'''

	{{Example: MLE for Exponential Distribution}}		{{Example: MLE for Exponential Distribution}}

Harry Guo: /* Maximum Likelihood Estimation for Exponential Distribution */

2012-02-09T18:15:22Z

Maximum Likelihood Estimation for Exponential Distribution

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 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. 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 equation with respect to the parameters, setting the resulting equations equal to zero, and solving simultaneously to determine the values of the parameter estimates. The log-likelihood functions and associated partial derivatives used to determine maximum likelihood estimates for the exponential distribution are covered in [[Appendix: Distribution Log-Likelihood Equations|Appendix]].
-''' MLE for Exponential Distribution'''
+'''Example 4:'''
-Using the data of Example 2 and assuming a two-parameter exponential distribution, estimate the parameters using the MLE method.
+{{Example: MLE for Exponential Distribution}}

Harry Guo: /* Example 4: MLE for Exponential Distribution */

2012-02-09T18:10:33Z

Example 4: MLE for Exponential Distribution

← Older revision		Revision as of 18:10, 9 February 2012
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	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. 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 equation with respect to the parameters, setting the resulting equations equal to zero, and solving simultaneously to determine the values of the parameter estimates. The log-likelihood functions and associated partial derivatives used to determine maximum likelihood estimates for the exponential distribution are covered in [[Appendix: Distribution Log-Likelihood Equations\|Appendix]].		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. 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 equation with respect to the parameters, setting the resulting equations equal to zero, and solving simultaneously to determine the values of the parameter estimates. The log-likelihood functions and associated partial derivatives used to determine maximum likelihood estimates for the exponential distribution are covered in [[Appendix: Distribution Log-Likelihood Equations\|Appendix]].

	~~====Example 4:~~ MLE for Exponential Distribution~~====~~		''' MLE for Exponential Distribution'''

	Using the data of Example 2 and assuming a two-parameter exponential distribution, estimate the parameters using the MLE method.		Using the data of Example 2 and assuming a two-parameter exponential distribution, estimate the parameters using the MLE method.

Harry Guo: /* Example 4: MLE for Exponential Distribution */

2012-02-08T22:40:49Z

Example 4: MLE for Exponential Distribution

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	Using the data of Example 2 and assuming a two-parameter exponential distribution, estimate the parameters using the MLE method.		Using the data of Example 2 and assuming a two-parameter exponential distribution, estimate the parameters using the MLE method.

	'''Solution to Example 4''''		'''Solution to Example 4'''

	In this example we have complete data only. The partial derivative of the log-likelihood function, <math>\Lambda ,</math> is given by:		In this example we have complete data only. The partial derivative of the log-likelihood function, <math>\Lambda ,</math> is given by:

Harry Guo: /* Solution to Example 4 */

2012-02-08T22:14:09Z

Solution to Example 4

← Older revision		Revision as of 22:14, 8 February 2012
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	Using the data of Example 2 and assuming a two-parameter exponential distribution, estimate the parameters using the MLE method.		Using the data of Example 2 and assuming a two-parameter exponential distribution, estimate the parameters using the MLE method.

	~~=====~~Solution to Example 4~~=====~~		'''Solution to Example 4''''

	In this example we have complete data only. The partial derivative of the log-likelihood function, <math>\Lambda ,</math> is given by:		In this example we have complete data only. The partial derivative of the log-likelihood function, <math>\Lambda ,</math> is given by:

Harry Guo: /* Maximum Likelihood Estimation for Exponential Distribution */

2012-02-07T23:37:52Z

Maximum Likelihood Estimation for Exponential Distribution

← Older revision		Revision as of 23:37, 7 February 2012
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	===Maximum Likelihood Estimation for Exponential Distribution===		===Maximum Likelihood Estimation for Exponential Distribution===
	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. 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 equation with respect to the parameters, setting the resulting equations equal to zero, and solving simultaneously to determine the values of the parameter estimates. The log-likelihood functions and associated partial derivatives used to determine maximum likelihood estimates for the exponential 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. 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 equation with respect to the parameters, setting the resulting equations equal to zero, and solving simultaneously to determine the values of the parameter estimates. The log-likelihood functions and associated partial derivatives used to determine maximum likelihood estimates for the exponential distribution are covered in [[Appendix: Distribution Log-Likelihood Equations\|Appendix]].

	====Example 4: MLE for Exponential Distribution====		====Example 4: MLE for Exponential Distribution====
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	Complete descriptions of the partial derivatives can be found in Appendix C. Recall that when using the MLE method for the exponential distribution, the value of <math>\gamma </math> is equal to that of the first failure time. The first failure occurred at 5 hours, thus <math>\gamma =5</math> hours<math>.</math> Substituting the values for <math>T</math> and <math>\gamma </math> we get:		Complete descriptions of the partial derivatives can be found in [[Appendix: Distribution Log-Likelihood Equations\|Appendix]]. Recall that when using the MLE method for the exponential distribution, the value of <math>\gamma </math> is equal to that of the first failure time. The first failure occurred at 5 hours, thus <math>\gamma =5</math> hours<math>.</math> Substituting the values for <math>T</math> and <math>\gamma </math> we get:

	::<math>\frac{14}{\hat{\lambda }}=560</math>		::<math>\frac{14}{\hat{\lambda }}=560</math>

Harry Guo: /* Example 4 */

2012-02-06T23:59:30Z

Example 4

← Older revision		Revision as of 23:59, 6 February 2012
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	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. 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 equation with respect to the parameters, setting the resulting equations equal to zero, and solving simultaneously to determine the values of the parameter estimates. The log-likelihood functions and associated partial derivatives used to determine maximum likelihood estimates for the exponential distribution are covered in Appendix C.		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. 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 equation with respect to the parameters, setting the resulting equations equal to zero, and solving simultaneously to determine the values of the parameter estimates. The log-likelihood functions and associated partial derivatives used to determine maximum likelihood estimates for the exponential distribution are covered in Appendix C.

	====Example 4====		====Example 4: MLE for Exponential Distribution====
	Using the data of Example 2 and assuming a two-parameter exponential distribution, estimate the parameters using the MLE method.		Using the data of Example 2 and assuming a two-parameter exponential distribution, estimate the parameters using the MLE method.

Harry Guo: /* Solution to Example 4 */

2012-02-06T23:58:29Z

Solution to Example 4

← Older revision		Revision as of 23:58, 6 February 2012
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	:Using Weibull++:		Using Weibull++:

	[[Image:weibullfolio1.png\|thumb\|center\|400px\|]]		[[Image:weibullfolio1.png\|thumb\|center\|400px\|]]

	:The probability plot is:		The probability plot is:

	[[Image:weibullfolioplot1.png\|thumb\|center\|400px\|]]		[[Image:weibullfolioplot1.png\|thumb\|center\|400px\|]]