Lognormal MLE Solution with Right Censored Data - Revision history

Kate Racaza at 16:18, 28 September 2015

2015-09-28T16:18:14Z

← Older revision		Revision as of 16:18, 28 September 2015
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	{{Reference Example}}		{{Reference Example}}

	This example ~~compares~~ the Lognormal MLE solution with Fisher matrix bound for right censored data.		This example validates the calculations for the Lognormal MLE solution with Fisher matrix bound for right censored data in Weibull++ standard folios.

Kate Racaza at 15:57, 9 June 2014

2014-06-09T15:57:33Z

← Older revision		Revision as of 15:57, 9 June 2014
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	{{Reference_Example_Heading1}}		{{Reference_Example_Heading1}}
	The data on page 199 of the book ''Statistical Methods for Reliability Data'' by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998 is used.		The data set on page 199 of the book ''Statistical Methods for Reliability Data'' by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998 is used.


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	* The MLE solution is <math>\hat{\mu} = 6.56, \ \hat{\sigma} = 0.543\,\!</math>.		* The MLE solution is <math>\hat{\mu} = 6.56, \ \hat{\sigma} = 0.543\,\!</math>.


	* The variance and covariance matrix is		* The variance and covariance matrix is
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	* The MLE solution and the variance/covariance matrix:		* The following picture shows the MLE solution and the variance/covariance matrix:

	[[Image:Lg_right_censored_matrix.png\|center]]		[[Image:Lg_right_censored_matrix.png\|center]]


	* The Fisher matrix bound for parameters:		* The Fisher matrix bound for parameters are:

	:For Ln-mu (using normal approximation of Eqn. 8.7 on page 187):		:For Ln-mu (using normal approximation of Eqn. 8.7 on page 187):

	::<math>[\hat{\mu}'_{L}, \hat{\mu}'_{U}] = \hat{\mu}' \pm z_{(1-\alpha ~~\setminus~~ 2)}se_{\hat{\mu}'}\,\!</math>		::<math>[\hat{\mu}'_{L}, \hat{\mu}'_{U}] = \hat{\mu}' \pm z_{(1-\alpha / 2)}se_{\hat{\mu}'}\,\!</math>


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	::<math>\begin{alignat}{2}		::<math>\begin{alignat}{2}
	[\hat{\mu}'_{L}, \hat{\mu}'_{U}] =& \hat{\mu}' \pm z_{(1-\alpha ~~\setminus~~ 2)}se_{\hat{\mu}'}\\		[\hat{\mu}'_{L}, \hat{\mu}'_{U}] =& \hat{\mu}' \pm z_{(1-\alpha /2)}se_{\hat{\mu}'}\\
	=& 6.564256 \pm 1.96 \times (0.0581)^{0.5}\\		=& 6.564256 \pm 1.96 \times (0.0581)^{0.5}\\
	=& [6.0918, 7.0366933]\\		=& [6.0918, 7.0366933]\\
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	:For Ln-Std (using log-normal approximation of Eqn. 8.8 on page 188):		:For Ln-Std (using log-normal approximation of Eqn. 8.8 on page 188):

	::<math>[\hat{\sigma}'_{L}, \hat{\sigma}'_{U}] = \hat{\sigma}'exp (\pm \frac{z_{(1-\alpha ~~\setminus~~ 2)}se_{\hat{\sigma'}}}{\hat{\sigma}'})\,\!</math>		::<math>[\hat{\sigma}'_{L}, \hat{\sigma}'_{U}] = \hat{\sigma}'exp (\pm \frac{z_{(1-\alpha /2)}se_{\hat{\sigma}'}}{\hat{\sigma}'})\,\!</math>


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	::<math>\begin{alignat}{2}		::<math>\begin{alignat}{2}
	[\hat{\sigma}'_{L}, \hat{\sigma}'_{U}] =& \hat{\sigma}'exp (\pm \frac{z_{(1-\alpha ~~\setminus~~ 2)}se_{\hat{\sigma'}}}{\hat{\sigma}'})\\		[\hat{\sigma}'_{L}, \hat{\sigma}'_{U}] =& \hat{\sigma}'exp (\pm \frac{z_{(1-\alpha /2)}se_{\hat{\sigma}'}}{\hat{\sigma}'})\\
	=& 0.534049 \times exp(\pm \tfrac{1.96\times 0.040562^{0.5}}{0.534049})\\		=& 0.534049 \times exp(\pm \tfrac{1.96\times 0.040562^{0.5}}{0.534049})\\
	=& [0.255, 1.118]\\		=& [0.255, 1.118]\\
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	:The results in Weibull++ ~~are~~:		:The following picture shows the results in Weibull++:

	[[Image:Lg_right_censored_bounds.png\|center]]		[[Image:Lg_right_censored_bounds.png\|center]]

Kate Racaza: Created page with '{{Reference Example}} This example compares the Lognormal MLE solution with Fisher matrix bound for right censored data. {{Reference_Example_Heading1}} The data on page 199 of…'

2014-06-04T20:39:35Z

Created page with '{{Reference Example}} This example compares the Lognormal MLE solution with Fisher matrix bound for right censored data. {{Reference_Example_Heading1}} The data on page 199 of…'

New page

{{Reference Example}}

This example compares the Lognormal MLE solution with Fisher matrix bound for right censored data.

{{Reference_Example_Heading1}}
The data on page 199 of the book ''Statistical Methods for Reliability Data'' by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998 is used.

{{Reference_Example_Heading2}}

{| {{table}}
!Number in State
!State F or S
!Time to Failure
|-
| 1||F||252
|-
| 1||F||315
|-
| 1||F||369
|-
| 1||F||403
|-
| 1||F||474
|-
| 15||S||500
|-
|}

{{Reference_Example_Heading3}}

* The MLE solution is <math>\hat{\mu} = 6.56, \ \hat{\sigma} = 0.543\,\!</math>.

* The variance and covariance matrix is
::<math>
\sum =\begin{bmatrix}
0.0581 & 0.0374 \\
0.0374 & 0.0405
\end{bmatrix}\,\!</math>

{{Reference_Example_Heading4}}

* The MLE solution and the variance/covariance matrix:

[[Image:Lg_right_censored_matrix.png|center]]

* The Fisher matrix bound for parameters:

:For Ln-mu (using normal approximation of Eqn. 8.7 on page 187):

::<math>[\hat{\mu}'_{L}, \hat{\mu}'_{U}] = \hat{\mu}' \pm z_{(1-\alpha \setminus 2)}se_{\hat{\mu}'}\,\!</math>

:For a confidence level of 0.95, it is:

::<math>\begin{alignat}{2}
[\hat{\mu}'_{L}, \hat{\mu}'_{U}] =& \hat{\mu}' \pm z_{(1-\alpha \setminus 2)}se_{\hat{\mu}'}\\
=& 6.564256 \pm 1.96 \times (0.0581)^{0.5}\\
=& [6.0918, 7.0366933]\\
\end{alignat}\,\!</math>

:For Ln-Std (using log-normal approximation of Eqn. 8.8 on page 188):

::<math>[\hat{\sigma}'_{L}, \hat{\sigma}'_{U}] = \hat{\sigma}'exp (\pm \frac{z_{(1-\alpha \setminus 2)}se_{\hat{\sigma'}}}{\hat{\sigma}'})\,\!</math>

:For confidence level of 0.95, it is:

::<math>\begin{alignat}{2}
[\hat{\sigma}'_{L}, \hat{\sigma}'_{U}] =& \hat{\sigma}'exp (\pm \frac{z_{(1-\alpha \setminus 2)}se_{\hat{\sigma'}}}{\hat{\sigma}'})\\
=& 0.534049 \times exp(\pm \tfrac{1.96\times 0.040562^{0.5}}{0.534049})\\
=& [0.255, 1.118]\\
\end{alignat} \,\!</math>

:The results in Weibull++ are:

[[Image:Lg_right_censored_bounds.png|center]]