Template:Weibull bayesian confidence bounds - Revision history

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

2012-08-13T02:13:54Z

Redirected page to The Weibull Distribution#Bayesian Confidence Bounds

← Older revision		Revision as of 02:13, 13 August 2012
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	~~== Bayesian Confidence Bounds ==~~		#REDIRECT [[The_Weibull_Distribution#Bayesian_Confidence_Bounds]]

	~~=== Bounds on Parameters ===~~

	~~Bayesian Bounds use non-informative prior distributions for both parameters. From Chapter~~ [[~~Confidence Bounds~~]], we know that if the prior distribution of <span class="texhtml">η</span> and <span class="texhtml">β</span> are independent, the posterior joint distribution of <span class="texhtml">η</span> and <span class="texhtml">β</span> can be written as:

	~~::<math> f(\eta ,\beta \|Data)= \dfrac{L(Data\|\eta ,\beta )\varphi (\eta )\varphi (\beta )}{\int_{0}^{\infty }\int_{0}^{\infty }L(Data\|\eta ,\beta )\varphi (\eta )\varphi (\beta )d\eta d\beta } </math>~~

	~~The marginal distribution of <span class="texhtml">η</span> is:~~

	~~::<math> f(\eta \|Data) =\int_{0}^{\infty }f(\eta ,\beta \|Data)d\beta =~~
	~~\dfrac{\int_{0}^{\infty }L(Data\|\eta ,\beta )\varphi (\eta )\varphi (\beta )d\beta }{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(Data\|\eta ,\beta )\varphi (\eta )\varphi (\beta )d\eta d\beta }~~
	~~</math>~~

	where: <math> \varphi (\beta )=\frac{1}{\beta } </math> is the non-informative prior of <span class="texhtml">β</span>. <math> \varphi (\eta )=\frac{1}{\eta } </math> is the non-informative prior of <span class="texhtml">η</span>. Using these non-informative prior distributions, <math>f(\eta\|Data)</math> can be rewritten as:

	::<math> f(\eta \|Data)=\dfrac{\int_{0}^{\infty }L(Data\|\eta ,\beta )\frac{1}{\beta } \frac{1}{\eta }d\beta }{\int_{0}^{\infty }\int_{0}^{\infty }L(Data\|\eta ,\beta )\frac{1}{\beta }\frac{1}{\eta }d\eta d\beta } </math>

	~~The one-sided upper bounds of <span class="texhtml">η</span> is:~~

	~~::<math> CL=P(\eta \leq \eta _{U})=\int_{0}^{\eta _{U}}f(\eta \|Data)d\eta </math>~~

	~~The one-sided lower bounds of <span class="texhtml">η</span> is:~~

	~~::<math> 1-CL=P(\eta \leq \eta _{L})=\int_{0}^{\eta _{L}}f(\eta \|Data)d\eta </math>~~

	~~The two-sided bounds of <span class="texhtml">η</span> is:~~

	~~::<math> CL=P(\eta _{L}\leq \eta \leq \eta _{U})=\int_{\eta _{L}}^{\eta _{U}}f(\eta \|Data)d\eta </math>~~

	~~Same method is used to obtain the bounds of <span class="texhtml">β</span>.~~

	~~=== Bounds on Reliability ===~~

	~~::<math> CL=\Pr (R\leq R_{U})=\Pr (\eta \leq T\exp (-\frac{\ln (-\ln R_{U})}{\beta })) </math>~~

	~~From the posterior distribution of <span class="texhtml">η,</span> we have:~~

	::<math> CL=\dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T\exp (-\dfrac{\ln (-\ln R_{U})}{\beta })}L(\beta ,\eta )\frac{1}{\beta }\frac{1}{\eta }d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\frac{1}{\beta }\frac{1}{\eta }d\eta d\beta } </math>

	The above equation is solved numerically for <span class="texhtml">''R''<sub>''U''</sub>.</span> The same method can be used to calculate the one sided lower bounds and two-sided bounds on reliability.

	~~=== Bounds on Time ===~~

	~~From Chapter [[Confidence Bounds]], we know that:~~

	~~::<math> CL=\Pr (T\leq T_{U})=\Pr (\eta \leq T_{U}\exp (-\frac{\ln (-\ln R)}{\beta })) </math>~~

	~~From the posterior distribution of <span class="texhtml">η</span>, we have:~~

	::<math> CL=\dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T_{U}\exp (-\dfrac{ \ln (-\ln R)}{\beta })}L(\beta ,\eta )\frac{1}{\beta }\frac{1}{\eta }d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\frac{1}{\beta }\frac{1}{\eta }d\eta d\beta } </math>

	~~The above equation is solved numerically for <span class="texhtml">''T''<sub>''U''</sub>.</span> The same method can be applied to calculate one sided lower bounds and two-sided bounds on time.~~

Harry Guo: /* Bounds on Time */

2012-02-10T22:59:05Z

Bounds on Time

← Older revision		Revision as of 22:59, 10 February 2012
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	=== Bounds on Time ===		=== Bounds on Time ===

	From [[Confidence Bounds~~\|Chapter 6~~]], we know that:		From Chapter [[Confidence Bounds]], we know that:

	::<math> CL=\Pr (T\leq T_{U})=\Pr (\eta \leq T_{U}\exp (-\frac{\ln (-\ln R)}{\beta })) </math>		::<math> CL=\Pr (T\leq T_{U})=\Pr (\eta \leq T_{U}\exp (-\frac{\ln (-\ln R)}{\beta })) </math>

Harry Guo: /* Bounds on Parameters */

2012-02-10T22:56:41Z

Bounds on Parameters

← Older revision		Revision as of 22:56, 10 February 2012
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	=== Bounds on Parameters ===		=== Bounds on Parameters ===

	Bayesian Bounds use non-informative prior distributions for both parameters. From [[Confidence Bounds~~\|Chapter 6~~]], we know that if the prior distribution of <span class="texhtml">η</span> and <span class="texhtml">β</span> are independent, the posterior joint distribution of <span class="texhtml">η</span> and <span class="texhtml">β</span> can be written as:		Bayesian Bounds use non-informative prior distributions for both parameters. From Chapter [[Confidence Bounds]], we know that if the prior distribution of <span class="texhtml">η</span> and <span class="texhtml">β</span> are independent, the posterior joint distribution of <span class="texhtml">η</span> and <span class="texhtml">β</span> can be written as:

	::<math> f(\eta ,\beta \|Data)= \dfrac{L(Data\|\eta ,\beta )\varphi (\eta )\varphi (\beta )}{\int_{0}^{\infty }\int_{0}^{\infty }L(Data\|\eta ,\beta )\varphi (\eta )\varphi (\beta )d\eta d\beta } </math>		::<math> f(\eta ,\beta \|Data)= \dfrac{L(Data\|\eta ,\beta )\varphi (\eta )\varphi (\beta )}{\int_{0}^{\infty }\int_{0}^{\infty }L(Data\|\eta ,\beta )\varphi (\eta )\varphi (\beta )d\eta d\beta } </math>

Harry Guo: /* Bounds on Time */

2012-02-09T23:08:05Z

Bounds on Time

← Older revision		Revision as of 23:08, 9 February 2012
Line 43:		Line 43:
	=== Bounds on Time ===		=== Bounds on Time ===

	From ~~Cha~~[[Confidence Bounds\|Chapter 6]], we know that:		From [[Confidence Bounds\|Chapter 6]], we know that:

	::<math> CL=\Pr (T\leq T_{U})=\Pr (\eta \leq T_{U}\exp (-\frac{\ln (-\ln R)}{\beta })) </math>		::<math> CL=\Pr (T\leq T_{U})=\Pr (\eta \leq T_{U}\exp (-\frac{\ln (-\ln R)}{\beta })) </math>

Harry Guo: /* Bounds on Time */

2012-02-09T23:07:54Z

Bounds on Time

← Older revision		Revision as of 23:07, 9 February 2012
Line 43:		Line 43:
	=== Bounds on Time ===		=== Bounds on Time ===

	From Chapter 5, we know that:		From Cha[[Confidence Bounds\|Chapter 6]], we know that:

	::<math> CL=\Pr (T\leq T_{U})=\Pr (\eta \leq T_{U}\exp (-\frac{\ln (-\ln R)}{\beta })) </math>		::<math> CL=\Pr (T\leq T_{U})=\Pr (\eta \leq T_{U}\exp (-\frac{\ln (-\ln R)}{\beta })) </math>

Harry Guo: /* Bounds on Time */

2012-02-09T23:07:26Z

Bounds on Time

← Older revision		Revision as of 23:07, 9 February 2012
Line 45:		Line 45:
	From Chapter 5, we know that:		From Chapter 5, we know that:

	::<math> CL=\Pr (T\leq T_{U})=\Pr (\eta \leq T_{U}\exp (-\frac{\ln (-\ln R)}{\beta })) </math> ~~From the posterior distribution of <span class="texhtml">η</span>, we have:~~		::<math> CL=\Pr (T\leq T_{U})=\Pr (\eta \leq T_{U}\exp (-\frac{\ln (-\ln R)}{\beta })) </math>

	::<~~math~~> CL=\dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T_{U}\exp (-\dfrac{ \ln (-\ln R)}{\beta })}L(\beta ,\eta )\frac{1}{\beta }\frac{1}{\eta }d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\frac{1}{\beta }\frac{1}{\eta }d\eta d\beta } EQNREF BayesCLT </~~math~~>		From the posterior distribution of <span class="texhtml">η</span>, we have:

	~~Eqn.~~ (~~EQNREF BayesCLT~~ ) is solved numerically for <span class="texhtml">''T''<sub>''U''</sub>.</span> The same method can be applied to calculate one sided lower bounds and two-sided bounds on time.		::<math> CL=\dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T_{U}\exp (-\dfrac{ \ln (-\ln R)}{\beta })}L(\beta ,\eta )\frac{1}{\beta }\frac{1}{\eta }d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\frac{1}{\beta }\frac{1}{\eta }d\eta d\beta } </math>

			The above equation is solved numerically for <span class="texhtml">''T''<sub>''U''</sub>.</span> The same method can be applied to calculate one sided lower bounds and two-sided bounds on time.

Harry Guo: /* Bounds on Reliability */

2012-02-09T23:06:58Z

Bounds on Reliability

← Older revision		Revision as of 23:06, 9 February 2012
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	=== Bounds on Reliability ===		=== Bounds on Reliability ===

	::<math> CL=\Pr (R\leq R_{U})=\Pr (\eta \leq T\exp (-\frac{\ln (-\ln R_{U})}{\beta })) </math> ~~From the posterior distribution of <span class="texhtml">η,</span> we have:~~		::<math> CL=\Pr (R\leq R_{U})=\Pr (\eta \leq T\exp (-\frac{\ln (-\ln R_{U})}{\beta })) </math>

	::<~~math~~> ~~CL=\dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T\exp (-\dfrac{\ln (-\ln R_{U})}{\beta })}L(\beta~~ ,~~\eta )\frac{1}{\beta }\frac{1}{\eta }d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\frac{1}{\beta }\frac{1}{\eta }d\eta d\beta } EQNREF BayesCLR~~ </~~math~~>		From the posterior distribution of <span class="texhtml">η,</span> we have:

	~~Eqn.~~ (~~EQNREF BayesCLR~~ ) is solved numerically for <span class="texhtml">''R''<sub>''U''</sub>.</span> The same method can be used to calculate the one sided lower bounds and two-sided bounds on reliability.		::<math> CL=\dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T\exp (-\dfrac{\ln (-\ln R_{U})}{\beta })}L(\beta ,\eta )\frac{1}{\beta }\frac{1}{\eta }d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\frac{1}{\beta }\frac{1}{\eta }d\eta d\beta } </math>

			The above equation is solved numerically for <span class="texhtml">''R''<sub>''U''</sub>.</span> The same method can be used to calculate the one sided lower bounds and two-sided bounds on reliability.

	=== Bounds on Time ===		=== Bounds on Time ===

Harry Guo: /* Bounds on Parameters */

2012-02-09T23:06:20Z

Bounds on Parameters

← Older revision		Revision as of 23:06, 9 February 2012
Line 23:		Line 23:
	The one-sided lower bounds of <span class="texhtml">η</span> is:		The one-sided lower bounds of <span class="texhtml">η</span> is:

	::<math> 1-CL=P(\eta \leq \eta _{L})=\int_{0}^{\eta _{L}}f(\eta \|Data)d\eta </math> ~~The two-sided bounds of <span class="texhtml">η</span> is:~~		::<math> 1-CL=P(\eta \leq \eta _{L})=\int_{0}^{\eta _{L}}f(\eta \|Data)d\eta </math>

	::<math> CL=P(\eta _{L}\leq \eta \leq \eta _{U})=\int_{\eta _{L}}^{\eta _{U}}f(\eta \|Data)d\eta </math> Same method is used to obtain the bounds of <span class="texhtml">β</span>.		The two-sided bounds of <span class="texhtml">η</span> is:

			::<math> CL=P(\eta _{L}\leq \eta \leq \eta _{U})=\int_{\eta _{L}}^{\eta _{U}}f(\eta \|Data)d\eta </math>

			Same method is used to obtain the bounds of <span class="texhtml">β</span>.

	=== Bounds on Reliability ===		=== Bounds on Reliability ===

Harry Guo: /* Bounds on Parameters */

2012-02-09T23:05:45Z

Bounds on Parameters

← Older revision		Revision as of 23:05, 9 February 2012
Line 3:		Line 3:
	=== Bounds on Parameters ===		=== Bounds on Parameters ===

	Bayesian Bounds use non-informative prior distributions for both parameters. From Chapter 5, we know that if the prior distribution of <span class="texhtml">η</span> and <span class="texhtml">β</span> are independent, the posterior joint distribution of <span class="texhtml">η</span> and <span class="texhtml">β</span> can be written as:		Bayesian Bounds use non-informative prior distributions for both parameters. From [[Confidence Bounds\|Chapter 6]], we know that if the prior distribution of <span class="texhtml">η</span> and <span class="texhtml">β</span> are independent, the posterior joint distribution of <span class="texhtml">η</span> and <span class="texhtml">β</span> can be written as:

	::<math> f(\eta ,\beta \|Data)= \dfrac{L(Data\|\eta ,\beta )\varphi (\eta )\varphi (\beta )}{\int_{0}^{\infty }\int_{0}^{\infty }L(Data\|\eta ,\beta )\varphi (\eta )\varphi (\beta )d\eta d\beta } </math>		::<math> f(\eta ,\beta \|Data)= \dfrac{L(Data\|\eta ,\beta )\varphi (\eta )\varphi (\beta )}{\int_{0}^{\infty }\int_{0}^{\infty }L(Data\|\eta ,\beta )\varphi (\eta )\varphi (\beta )d\eta d\beta } </math>
Line 26:		Line 26:

	::<math> CL=P(\eta _{L}\leq \eta \leq \eta _{U})=\int_{\eta _{L}}^{\eta _{U}}f(\eta \|Data)d\eta </math> Same method is used to obtain the bounds of <span class="texhtml">β</span>.		::<math> CL=P(\eta _{L}\leq \eta \leq \eta _{U})=\int_{\eta _{L}}^{\eta _{U}}f(\eta \|Data)d\eta </math> Same method is used to obtain the bounds of <span class="texhtml">β</span>.


	=== Bounds on Reliability ===		=== Bounds on Reliability ===

Harry Guo: /* Bayesian Confidence Bounds */

2012-02-09T21:54:01Z

Bayesian Confidence Bounds

@@ Line 27: / Line 27: @@
 ::<math> CL=P(\eta _{L}\leq \eta \leq \eta _{U})=\int_{\eta _{L}}^{\eta _{U}}f(\eta |Data)d\eta </math> Same method is used to obtain the bounds of <span class="texhtml">β</span>.
-=== Bounds on Time (Type 1) ===
 From Chapter 5, we know that:
@@ Line 36: / Line 45: @@
 Eqn. (EQNREF BayesCLT ) is solved numerically for <span class="texhtml">''T''<sub>''U''</sub>.</span> The same method can be applied to calculate one sided lower bounds and two-sided bounds on time.