Template:Weibull bayesian confidence bounds: Difference between revisions

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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.

Latest revision as of 02:13, 13 August 2012

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