Template:Lognormal distribution Likelihood ratio confidence bounds: Difference between revisions

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===Likelihood Ratio Confidence Bounds===
#REDIRECT [[The_Lognormal_Distribution#Likelihood_Ratio_Confidence_Bounds]]
 
====Bounds on Parameters====
As covered in Chapter [[Parameter Estimation]], the likelihood confidence bounds are calculated by finding values for  <math>{{\theta }_{1}}</math>  and  <math>{{\theta }_{2}}</math>  that satisfy:
 
::<math>-2\cdot \text{ln}\left( \frac{L({{\theta }_{1}},{{\theta }_{2}})}{L({{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}})} \right)=\chi _{\alpha ;1}^{2}</math>
 
This equation can be rewritten as:
 
::<math>L({{\theta }_{1}},{{\theta }_{2}})=L({{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}})\cdot {{e}^{\tfrac{-\chi _{\alpha ;1}^{2}}{2}}}</math>
 
For complete data, the likelihood formula for the normal distribution is given by:
 
::<math>L({\mu }',{{\sigma' }})=\underset{i=1}{\overset{N}{\mathop \prod }}\,f({{x}_{i}};{\mu }',{{\sigma' }})=\underset{i=1}{\overset{N}{\mathop \prod }}\,\frac{1}{{{x}_{i}}\cdot {{\sigma' }}\cdot \sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}({{x}_{i}})-{\mu }'}{{{\sigma'}}} \right)}^{2}}}}</math>
 
where the  <math>{{x}_{i}}</math>  values represent the original time-to-failure data.  For a given value of  <math>\alpha </math> , values for  <math>{\mu }'</math>  and  <math>{{\sigma' }}</math>  can be found which represent the maximum and minimum values that satisfy likelihood ratio equation. These represent the confidence bounds for the parameters at a confidence level  <math>\delta ,</math>  where  <math>\alpha =\delta </math>  for two-sided bounds and  <math>\alpha =2\delta -1</math>  for one-sided.
 
 
'''Example 5:'''
{{Example: Lognormal Distribution Likelihood Ratio Bound (Parameters)}}
 
====Bounds on Time and Reliability====
In order to calculate the bounds on a time estimate for a given reliability, or on a reliability estimate for a given time, the likelihood function needs to be rewritten in terms of one parameter and time/reliability, so that the maximum and minimum values of the time can be observed as the parameter is varied. This can be accomplished by substituting a form of the normal reliability equation into the likelihood function. The normal reliability equation can be written as:
 
::<math>R=1-\Phi \left( \frac{\text{ln}(t)-{\mu }'}{{{\sigma'}}} \right)</math>
 
This can be rearranged to the form:
 
::<math>{\mu }'=\text{ln}(t)-{{\sigma'}}\cdot {{\Phi }^{-1}}(1-R)</math>
 
where  <math>{{\Phi }^{-1}}</math>  is the inverse standard normal. This equation can now be substituted into likelihood function to produce a likelihood equation in terms of  <math>{{\sigma'}},</math>  <math>t</math>  and  <math>R\ \ :</math> 
 
::<math>L({{\sigma'}},t/R)=\underset{i=1}{\overset{N}{\mathop \prod }}\,\frac{1}{{{x}_{i}}\cdot {{\sigma'}}\cdot \sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}({{x}_{i}})-\left( \text{ln}(t)-{{\sigma'}}\cdot {{\Phi }^{-1}}(1-R) \right)}{{{\sigma'}}} \right)}^{2}}}}</math>
 
The unknown variable  <math>t/R</math>  depends on what type of bounds are being determined.  If one is trying to determine the bounds on time for a given reliability, then  <math>R</math>  is a known constant and  <math>t</math>  is the unknown variable. Conversely, if one is trying to determine the bounds on reliability for a given time, then  <math>t</math>  is a known constant and  <math>R</math>  is the unknown variable. Either way, the above equation can be used to solve the likelihood ratio equation for the values of interest.
 
'''Example 6:'''
{{Example: Lognormal Distribution Likelihood Ratio Bound (Time)}}
 
 
'''Example 7:'''
{{Example: Lognormal Distribution Likelihood Ratio Bound (Reliability)}}

Latest revision as of 06:06, 13 August 2012

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