Template:Mixed weibull confidence bounds - Revision history

Richard House: Redirected page to The Mixed Weibull Distribution#Mixed Weibull Confidence Bounds

2012-08-15T00:41:40Z

Redirected page to The Mixed Weibull Distribution#Mixed Weibull Confidence Bounds

← Older revision		Revision as of 00:41, 15 August 2012
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	~~===Mixed Weibull Confidence Bounds===~~		#REDIRECT [[The_Mixed_Weibull_Distribution#Mixed_Weibull_Confidence_Bounds]]
	~~In Weibull++, two methods are available for estimating the confidence bounds for the mixed Weibull distribution. The first method is the beta binomial, described in~~ [[~~Confidence Bounds~~]] ~~chapter. The second method is the Fisher matrix confidence bounds.~~
	For the Fisher matrix bounds, the methodology is the same as described in [[Confidence Bounds]] chapter. The variance/covariance matrix for the mixed Weibull is a <math>(3\cdot S-1)\times (3\cdot S-1)</math> matrix, where <math>S</math> is the number of subpopulations. Bounds on the parameters, reliability and time are estimated using the same transformations and methods that were used for the [[The Weibull Distribution]] chapter. Note, however, that in addition to the Weibull parameters, the bounds on the subpopulation portions are obtained as well. The bounds on the portions are estimated by:

	~~::<math>\begin{align}~~
	~~& {{\rho }_{U}}= & \frac{{\hat{\rho }}}{\hat{\rho }+(1-\hat{\rho }){{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\rho })}}{\hat{\rho }(1-\hat{\rho })}}}} \\~~
	~~& & \\~~
	~~& {{\rho }_{L}}= & \frac{{\hat{\rho }}}{\hat{\rho }+(1-\hat{\rho }){{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\rho })}}{\hat{\rho }(1-\hat{\rho })}}}}~~
	~~\end{align}</math>~~


	~~where <math>Var(\widehat{\rho })</math> is obtained from the variance/covariance matrix.~~
	When using the Fisher matrix bounds method, problems can occur on the transition points of the distribution, and in particular on the Type 1 confidence bounds (bounds on time). The problems (i.e. the departure from the expected monotonic behavior) occur when the transition region between two subpopulations becomes a ``saddle'' (i.e. the probability line is almost parallel to the time axis on a probability plot). In this case, the bounds on time approach infinity. This behavior is more frequently encountered with smaller sample sizes. The physical interpretation is that there is insufficient data to support any inferences when in this region.

	This is graphically illustrated in the following figure. In this plot it can be seen that there are no data points between the last point of the first subpopulation and the first point of the second subpopulation, thus the uncertainty is high, as described by the mathematical model.

	~~[[Image:WBprobabilityweibullplot.png\|center\|250px\| ]]~~

	~~Beta binomial bounds can be used instead in these cases, especially when estimations are to be obtained close to these regions.~~

Nicolette Young: /* Mixed Weibull Confidence Bounds */

2012-04-25T21:22:52Z

Mixed Weibull Confidence Bounds

← Older revision		Revision as of 21:22, 25 April 2012
Line 15:		Line 15:
	This is graphically illustrated in the following figure. In this plot it can be seen that there are no data points between the last point of the first subpopulation and the first point of the second subpopulation, thus the uncertainty is high, as described by the mathematical model.		This is graphically illustrated in the following figure. In this plot it can be seen that there are no data points between the last point of the first subpopulation and the first point of the second subpopulation, thus the uncertainty is high, as described by the mathematical model.

	[[Image:WBprobabilityweibullplot.png\|center\|~~500px~~\| ]]		[[Image:WBprobabilityweibullplot.png\|center\|250px\| ]]

	Beta binomial bounds can be used instead in these cases, especially when estimations are to be obtained close to these regions.		Beta binomial bounds can be used instead in these cases, especially when estimations are to be obtained close to these regions.

Nicolette Young: /* Mixed Weibull Confidence Bounds */

2012-03-14T19:02:56Z

Mixed Weibull Confidence Bounds

← Older revision		Revision as of 19:02, 14 March 2012
Line 15:		Line 15:
	This is graphically illustrated in the following figure. In this plot it can be seen that there are no data points between the last point of the first subpopulation and the first point of the second subpopulation, thus the uncertainty is high, as described by the mathematical model.		This is graphically illustrated in the following figure. In this plot it can be seen that there are no data points between the last point of the first subpopulation and the first point of the second subpopulation, thus the uncertainty is high, as described by the mathematical model.

	[[Image:~~ldachp10fig1~~.~~gif\|thumb~~\|center\|500px\| ]]		[[Image:WBprobabilityweibullplot.png\|center\|500px\| ]]

	Beta binomial bounds can be used instead in these cases, especially when estimations are to be obtained close to these regions.		Beta binomial bounds can be used instead in these cases, especially when estimations are to be obtained close to these regions.

Harry Guo at 18:15, 14 February 2012

2012-02-14T18:15:29Z

← Older revision		Revision as of 18:15, 14 February 2012
Line 1:		Line 1:
	===Mixed Weibull Confidence Bounds===		===Mixed Weibull Confidence Bounds===
	In Weibull++, two methods are available for estimating the confidence bounds for the mixed Weibull distribution. The first method is the beta binomial, described in [[Confidence Bounds]] chapter. The second method is the Fisher matrix confidence bounds.		In Weibull++, two methods are available for estimating the confidence bounds for the mixed Weibull distribution. The first method is the beta binomial, described in [[Confidence Bounds]] chapter. The second method is the Fisher matrix confidence bounds.
	For the Fisher matrix bounds, the methodology is the same as described in [[Confidence Bounds]] chapter. The variance/covariance matrix for the mixed Weibull is a <math>(3\cdot S-1)\times (3\cdot S-1)</math> matrix, where <math>S</math> is the number of subpopulations. Bounds on the parameters, reliability and time are estimated using the same transformations and methods that were used for the [[The Weibull ~~distribution~~]] chapter. Note, however, that in addition to the Weibull parameters, the bounds on the subpopulation portions are obtained as well. The bounds on the portions are estimated by:		For the Fisher matrix bounds, the methodology is the same as described in [[Confidence Bounds]] chapter. The variance/covariance matrix for the mixed Weibull is a <math>(3\cdot S-1)\times (3\cdot S-1)</math> matrix, where <math>S</math> is the number of subpopulations. Bounds on the parameters, reliability and time are estimated using the same transformations and methods that were used for the [[The Weibull Distribution]] chapter. Note, however, that in addition to the Weibull parameters, the bounds on the subpopulation portions are obtained as well. The bounds on the portions are estimated by:

	::<math>\begin{align}		::<math>\begin{align}

Harry Guo: /* Mixed Weibull Confidence Bounds */

2012-02-14T18:14:52Z

Mixed Weibull Confidence Bounds

← Older revision		Revision as of 18:14, 14 February 2012
Line 1:		Line 1:
	===Mixed Weibull Confidence Bounds===		===Mixed Weibull Confidence Bounds===
	In Weibull++, two methods are available for estimating the confidence bounds for the mixed Weibull distribution. The first method is the beta binomial, described in ~~Chapter 5~~. The second method is the Fisher matrix confidence bounds.		In Weibull++, two methods are available for estimating the confidence bounds for the mixed Weibull distribution. The first method is the beta binomial, described in [[Confidence Bounds]] chapter. The second method is the Fisher matrix confidence bounds.
	For the Fisher matrix bounds, the methodology is the same as described in ~~Chapter 5~~. The variance/covariance matrix for the mixed Weibull is a <math>(3\cdot S-1)\times (3\cdot S-1)</math> matrix, where <math>S</math> is the number of subpopulations. Bounds on the parameters, reliability and time are estimated using the same transformations and methods that were used for the Weibull distribution ~~(Chapter 6)~~. Note, however, that in addition to the Weibull parameters, the bounds on the subpopulation portions are obtained as well. The bounds on the portions are estimated by:		For the Fisher matrix bounds, the methodology is the same as described in [[Confidence Bounds]] chapter. The variance/covariance matrix for the mixed Weibull is a <math>(3\cdot S-1)\times (3\cdot S-1)</math> matrix, where <math>S</math> is the number of subpopulations. Bounds on the parameters, reliability and time are estimated using the same transformations and methods that were used for the [[The Weibull distribution]] chapter. Note, however, that in addition to the Weibull parameters, the bounds on the subpopulation portions are obtained as well. The bounds on the portions are estimated by:

	::<math>\begin{align}		::<math>\begin{align}

Nicolette Young: Created page with '===Mixed Weibull Confidence Bounds=== In Weibull++, two methods are available for estimating the confidence bounds for the mixed Weibull distribution. The first method is the bet…'

2012-01-04T18:42:48Z

Created page with '===Mixed Weibull Confidence Bounds=== In Weibull++, two methods are available for estimating the confidence bounds for the mixed Weibull distribution. The first method is the bet…'

New page

===Mixed Weibull Confidence Bounds===
In Weibull++, two methods are available for estimating the confidence bounds for the mixed Weibull distribution. The first method is the beta binomial, described in Chapter 5. The second method is the Fisher matrix confidence bounds.
For the Fisher matrix bounds, the methodology is the same as described in Chapter 5. The variance/covariance matrix for the mixed Weibull is a <math>(3\cdot S-1)\times (3\cdot S-1)</math> matrix, where <math>S</math> is the number of subpopulations. Bounds on the parameters, reliability and time are estimated using the same transformations and methods that were used for the Weibull distribution (Chapter 6). Note, however, that in addition to the Weibull parameters, the bounds on the subpopulation portions are obtained as well. The bounds on the portions are estimated by:

::<math>\begin{align}
& {{\rho }_{U}}= & \frac{{\hat{\rho }}}{\hat{\rho }+(1-\hat{\rho }){{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\rho })}}{\hat{\rho }(1-\hat{\rho })}}}} \\
& & \\
& {{\rho }_{L}}= & \frac{{\hat{\rho }}}{\hat{\rho }+(1-\hat{\rho }){{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\rho })}}{\hat{\rho }(1-\hat{\rho })}}}}
\end{align}</math>

where <math>Var(\widehat{\rho })</math> is obtained from the variance/covariance matrix.
When using the Fisher matrix bounds method, problems can occur on the transition points of the distribution, and in particular on the Type 1 confidence bounds (bounds on time). The problems (i.e. the departure from the expected monotonic behavior) occur when the transition region between two subpopulations becomes a ``saddle'' (i.e. the probability line is almost parallel to the time axis on a probability plot). In this case, the bounds on time approach infinity. This behavior is more frequently encountered with smaller sample sizes. The physical interpretation is that there is insufficient data to support any inferences when in this region.

This is graphically illustrated in the following figure. In this plot it can be seen that there are no data points between the last point of the first subpopulation and the first point of the second subpopulation, thus the uncertainty is high, as described by the mathematical model.

[[Image:ldachp10fig1.gif|thumb|center|500px| ]]

Beta binomial bounds can be used instead in these cases, especially when estimations are to be obtained close to these regions.