Template:Weibull likelihood ratio confidence bounds - Revision history

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

2012-08-13T02:12:26Z

Redirected page to The Weibull Distribution#Likelihood Ratio Confidence Bounds

← Older revision		Revision as of 02:12, 13 August 2012
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	~~== Likelihood Ratio Confidence Bounds ==~~		#REDIRECT [[The_Weibull_Distribution#Likelihood_Ratio_Confidence_Bounds]]

	~~As covered in Chapter~~ [[~~Confidence Bounds~~]]~~, the likelihood confidence bounds are calculated by finding values for <span class="texhtml">θ<sub>1</sub></span> and <span class="texhtml">θ<sub>2</sub></span> that satisfy:~~

	~~::<math> -2\cdot \text{ln}\left( \frac{L(\theta _{1},\theta _{2})}{L(\hat{\theta }_{1}, \hat{\theta }_{2})}\right) =\chi _{\alpha ;1}^{2} </math>~~

	~~This equation can be rewritten as:~~

	~~::<math> L(\theta _{1},\theta _{2})=L(\hat{\theta }_{1},\hat{\theta } _{2})\cdot e^{\frac{-\chi _{\alpha ;1}^{2}}{2}} </math>~~

	~~For complete data, the likelihood function for the Weibull distribution is given by:~~

	::<math> L(\beta ,\eta )=\prod_{i=1}^{N}f(x_{i};\beta ,\eta )=\prod_{i=1}^{N}\frac{ \beta }{\eta }\cdot \left( \frac{x_{i}}{\eta }\right) ^{\beta -1}\cdot e^{-\left( \frac{x_{i}}{\eta }\right) ^{\beta }} </math>

	For a given value of <span class="texhtml">α</span>, values for <span class="texhtml">β</span> and <span class="texhtml">η</span> can be found which represent the maximum and minimum values that satisfy the above equation. These represent the confidence bounds for the parameters at a confidence level <span class="texhtml">δ</span>, where <span class="texhtml">α = δ</span> for two-sided bounds and <span class="texhtml">α = 2δ − 1</span> for one-sided.

	Similarly, the bounds on time and reliability can be found by substituting the Weibull reliability equation into the likelihood function so that it is in terms of <span class="texhtml">β</span> and time or reliability, as discussed in Chapter [[Confidence Bounds]]. The likelihood ratio equation used to solve for bounds on time (Type 1) is:


	::<math> L(\beta ,t)=\prod_{i=1}^{N}\frac{\beta }{\left( \frac{t}{(-\text{ln}(R))^{ \frac{1}{\beta }}}\right) }\cdot \left( \frac{x_{i}}{\left( \frac{t}{(-\text{ ln}(R))^{\frac{1}{\beta }}}\right) }\right) ^{\beta -1}\cdot \text{exp}\left[ -\left( \frac{x_{i}}{\left( \frac{t}{(-\text{ln}(R))^{\frac{1}{\beta }}} \right) }\right) ^{\beta }\right] </math>

	~~The likelihood ratio equation used to solve for bounds on reliability (Type 2) is:~~

	::<math> L(\beta ,R)=\prod_{i=1}^{N}\frac{\beta }{\left( \frac{t}{(-\text{ln}(R))^{ \frac{1}{\beta }}}\right) }\cdot \left( \frac{x_{i}}{\left( \frac{t}{(-\text{ ln}(R))^{\frac{1}{\beta }}}\right) }\right) ^{\beta -1}\cdot \text{exp}\left[ -\left( \frac{x_{i}}{\left( \frac{t}{(-\text{ln}(R))^{\frac{1}{\beta }}} \right) }\right) ^{\beta }\right] </math>

Harry Guo at 22:54, 10 February 2012

2012-02-10T22:54:33Z

← Older revision		Revision as of 22:54, 10 February 2012
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	For a given value of <span class="texhtml">α</span>, values for <span class="texhtml">β</span> and <span class="texhtml">η</span> can be found which represent the maximum and minimum values that satisfy the above equation. These represent the confidence bounds for the parameters at a confidence level <span class="texhtml">δ</span>, where <span class="texhtml">α = δ</span> for two-sided bounds and <span class="texhtml">α = 2δ − 1</span> for one-sided.		For a given value of <span class="texhtml">α</span>, values for <span class="texhtml">β</span> and <span class="texhtml">η</span> can be found which represent the maximum and minimum values that satisfy the above equation. These represent the confidence bounds for the parameters at a confidence level <span class="texhtml">δ</span>, where <span class="texhtml">α = δ</span> for two-sided bounds and <span class="texhtml">α = 2δ − 1</span> for one-sided.

	Similarly, the bounds on time and reliability can be found by substituting the Weibull reliability equation into the likelihood function so that it is in terms of <span class="texhtml">β</span> and time or reliability, as discussed in [[Confidence Bounds~~\|Chapter 6~~]]. The likelihood ratio equation used to solve for bounds on time (Type 1) is:		Similarly, the bounds on time and reliability can be found by substituting the Weibull reliability equation into the likelihood function so that it is in terms of <span class="texhtml">β</span> and time or reliability, as discussed in Chapter [[Confidence Bounds]]. The likelihood ratio equation used to solve for bounds on time (Type 1) is:

Harry Guo: /* Likelihood Ratio Confidence Bounds */

2012-02-10T22:53:54Z

Likelihood Ratio Confidence Bounds

← Older revision		Revision as of 22:53, 10 February 2012
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	== Likelihood Ratio Confidence Bounds ==		== Likelihood Ratio Confidence Bounds ==

	As covered in [[Confidence Bounds~~\|Chapter 6~~]], the likelihood confidence bounds are calculated by finding values for <span class="texhtml">θ<sub>1</sub></span> and <span class="texhtml">θ<sub>2</sub></span> that satisfy:		As covered in Chapter [[Confidence Bounds]], the likelihood confidence bounds are calculated by finding values for <span class="texhtml">θ<sub>1</sub></span> and <span class="texhtml">θ<sub>2</sub></span> that satisfy:

	::<math> -2\cdot \text{ln}\left( \frac{L(\theta _{1},\theta _{2})}{L(\hat{\theta }_{1}, \hat{\theta }_{2})}\right) =\chi _{\alpha ;1}^{2} </math>		::<math> -2\cdot \text{ln}\left( \frac{L(\theta _{1},\theta _{2})}{L(\hat{\theta }_{1}, \hat{\theta }_{2})}\right) =\chi _{\alpha ;1}^{2} </math>

Harry Guo at 23:04, 9 February 2012

2012-02-09T23:04:53Z

← Older revision		Revision as of 23:04, 9 February 2012
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	This equation can be rewritten as:		This equation can be rewritten as:


	::<math> L(\theta _{1},\theta _{2})=L(\hat{\theta }_{1},\hat{\theta } _{2})\cdot e^{\frac{-\chi _{\alpha ;1}^{2}}{2}} </math>		::<math> L(\theta _{1},\theta _{2})=L(\hat{\theta }_{1},\hat{\theta } _{2})\cdot e^{\frac{-\chi _{\alpha ;1}^{2}}{2}} </math>
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	For complete data, the likelihood function for the Weibull distribution is given by:		For complete data, the likelihood function for the Weibull distribution is given by:


	::<math> L(\beta ,\eta )=\prod_{i=1}^{N}f(x_{i};\beta ,\eta )=\prod_{i=1}^{N}\frac{ \beta }{\eta }\cdot \left( \frac{x_{i}}{\eta }\right) ^{\beta -1}\cdot e^{-\left( \frac{x_{i}}{\eta }\right) ^{\beta }} </math>		::<math> L(\beta ,\eta )=\prod_{i=1}^{N}f(x_{i};\beta ,\eta )=\prod_{i=1}^{N}\frac{ \beta }{\eta }\cdot \left( \frac{x_{i}}{\eta }\right) ^{\beta -1}\cdot e^{-\left( \frac{x_{i}}{\eta }\right) ^{\beta }} </math>

Harry Guo: /* Likelihood Ratio Confidence Bounds */

2012-02-09T23:04:41Z

Likelihood Ratio Confidence Bounds

← Older revision		Revision as of 23:04, 9 February 2012
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	== Likelihood Ratio Confidence Bounds ==		== Likelihood Ratio Confidence Bounds ==

	As covered in Chapter 5, the likelihood confidence bounds are calculated by finding values for <span class="texhtml">θ<sub>1</sub></span> and <span class="texhtml">θ<sub>2</sub></span> that satisfy:		As covered in [[Confidence Bounds\|Chapter 6]], the likelihood confidence bounds are calculated by finding values for <span class="texhtml">θ<sub>1</sub></span> and <span class="texhtml">θ<sub>2</sub></span> that satisfy:

	::<math> -2\cdot \text{ln}\left( \frac{L(\theta _{1},\theta _{2})}{L(\hat{\theta }_{1}, \hat{\theta }_{2})}\right) =\chi _{\alpha ;1}^{2} ~~EQNREF lratio2~~ </math>		::<math> -2\cdot \text{ln}\left( \frac{L(\theta _{1},\theta _{2})}{L(\hat{\theta }_{1}, \hat{\theta }_{2})}\right) =\chi _{\alpha ;1}^{2} </math>

	This equation can be rewritten as:		This equation can be rewritten as:

	~~::<math> L(\theta _{1},\theta _{2})=L(\hat{\theta }_{1},\hat{\theta } _{2})\cdot e^{\frac{-\chi _{\alpha ;1}^{2}}{2}} EQNREF lratio3 </math>~~

	For complete data, the likelihood function for the Weibull distribution is given by:		::<math> L(\theta _{1},\theta _{2})=L(\hat{\theta }_{1},\hat{\theta } _{2})\cdot e^{\frac{-\chi _{\alpha ;1}^{2}}{2}} </math>

			For complete data, the likelihood function for the Weibull distribution is given by:


	::<math> L(\beta ,\eta )=\prod_{i=1}^{N}f(x_{i};\beta ,\eta )=\prod_{i=1}^{N}\frac{ \beta }{\eta }\cdot \left( \frac{x_{i}}{\eta }\right) ^{\beta -1}\cdot e^{-\left( \frac{x_{i}}{\eta }\right) ^{\beta }} </math>		::<math> L(\beta ,\eta )=\prod_{i=1}^{N}f(x_{i};\beta ,\eta )=\prod_{i=1}^{N}\frac{ \beta }{\eta }\cdot \left( \frac{x_{i}}{\eta }\right) ^{\beta -1}\cdot e^{-\left( \frac{x_{i}}{\eta }\right) ^{\beta }} </math>

	For a given value of <span class="texhtml">α</span>, values for <span class="texhtml">β</span> and <span class="texhtml">η</span> can be found which represent the maximum and minimum values that satisfy ~~Eqn. (\ref {lratio3})~~. These represent the confidence bounds for the parameters at a confidence level <span class="texhtml">δ</span>, where <span class="texhtml">α = δ</span> for two-sided bounds and <span class="texhtml">α = 2δ − 1</span> for one-sided.		For a given value of <span class="texhtml">α</span>, values for <span class="texhtml">β</span> and <span class="texhtml">η</span> can be found which represent the maximum and minimum values that satisfy the above equation. These represent the confidence bounds for the parameters at a confidence level <span class="texhtml">δ</span>, where <span class="texhtml">α = δ</span> for two-sided bounds and <span class="texhtml">α = 2δ − 1</span> for one-sided.

	Similarly, the bounds on time and reliability can be found by substituting the Weibull reliability equation into the likelihood function so that it is in terms of <span class="texhtml">β</span> and time or reliability, as discussed in Chapter 5. The likelihood ratio equation used to solve for bounds on time (Type 1) is:		Similarly, the bounds on time and reliability can be found by substituting the Weibull reliability equation into the likelihood function so that it is in terms of <span class="texhtml">β</span> and time or reliability, as discussed in [[Confidence Bounds\|Chapter 6]]. The likelihood ratio equation used to solve for bounds on time (Type 1) is:

Nicolette Young: Created page with '== Likelihood Ratio Confidence Bounds == As covered in Chapter 5, the likelihood confidence bounds are calculated by finding values for $θ1 2012-01-04T16:52:51Z Created page with '== Likelihood Ratio Confidence Bounds == As covered in Chapter 5, the likelihood confidence bounds are calculated by finding values for θ1</spa…' New page== Likelihood Ratio Confidence Bounds == As covered in Chapter 5, the likelihood confidence bounds are calculated by finding values for θ1 and θ2 that satisfy: ::<math> -2\cdot \text{ln}\left( \frac{L(\theta _{1},\theta _{2})}{L(\hat{\theta }_{1}, \hat{\theta }_{2})}\right) =\chi _{\alpha ;1}^{2} EQNREF lratio2 </math> This equation can be rewritten as: ::<math> L(\theta _{1},\theta _{2})=L(\hat{\theta }_{1},\hat{\theta } _{2})\cdot e^{\frac{-\chi _{\alpha ;1}^{2}}{2}} EQNREF lratio3 </math> For complete data, the likelihood function for the Weibull distribution is given by: ::<math> L(\beta ,\eta )=\prod_{i=1}^{N}f(x_{i};\beta ,\eta )=\prod_{i=1}^{N}\frac{ \beta }{\eta }\cdot \left( \frac{x_{i}}{\eta }\right) ^{\beta -1}\cdot e^{-\left( \frac{x_{i}}{\eta }\right) ^{\beta }} </math> For a given value of α, values for β and η can be found which represent the maximum and minimum values that satisfy Eqn. (\ref {lratio3}). These represent the confidence bounds for the parameters at a confidence level δ, where α = δ for two-sided bounds and α = 2δ − 1 for one-sided. Similarly, the bounds on time and reliability can be found by substituting the Weibull reliability equation into the likelihood function so that it is in terms of β and time or reliability, as discussed in Chapter 5. The likelihood ratio equation used to solve for bounds on time (Type 1) is: ::<math> L(\beta ,t)=\prod_{i=1}^{N}\frac{\beta }{\left( \frac{t}{(-\text{ln}(R))^{ \frac{1}{\beta }}}\right) }\cdot \left( \frac{x_{i}}{\left( \frac{t}{(-\text{ ln}(R))^{\frac{1}{\beta }}}\right) }\right) ^{\beta -1}\cdot \text{exp}\left[ -\left( \frac{x_{i}}{\left( \frac{t}{(-\text{ln}(R))^{\frac{1}{\beta }}} \right) }\right) ^{\beta }\right] </math> The likelihood ratio equation used to solve for bounds on reliability (Type 2) is: ::<math> L(\beta ,R)=\prod_{i=1}^{N}\frac{\beta }{\left( \frac{t}{(-\text{ln}(R))^{ \frac{1}{\beta }}}\right) }\cdot \left( \frac{x_{i}}{\left( \frac{t}{(-\text{ ln}(R))^{\frac{1}{\beta }}}\right) }\right) ^{\beta -1}\cdot \text{exp}\left[ -\left( \frac{x_{i}}{\left( \frac{t}{(-\text{ln}(R))^{\frac{1}{\beta }}} \right) }\right) ^{\beta }\right] </math>$