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| ===Likelihood Ratio Confidence Bounds===
| | #REDIRECT [[The_Lognormal_Distribution#Likelihood_Ratio_Confidence_Bounds]] |
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| ====Bounds on Parameters====
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| 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:
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| ::<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>
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| This equation can be rewritten as:
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| ::<math>L({{\theta }_{1}},{{\theta }_{2}})=L({{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}})\cdot {{e}^{\tfrac{-\chi _{\alpha ;1}^{2}}{2}}}</math>
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| For complete data, the likelihood formula for the normal distribution is given by:
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| ::<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>
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| 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.
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| '''Example 5:'''
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| {{Example: Lognormal Distribution Likelihood Ratio Bound (Parameters)}}
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| ====Bounds on Time and Reliability====
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| 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:
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| ::<math>R=1-\Phi \left( \frac{\text{ln}(t)-{\mu }'}{{{\sigma }_{{{T}'}}}} \right)</math>
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| This can be rearranged to the form:
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| ::<math>{\mu }'=\text{ln}(t)-{{\sigma }_{{{T}'}}}\cdot {{\Phi }^{-1}}(1-R)</math>
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| where <math>{{\Phi }^{-1}}</math> is the inverse standard normal. This equation can now be substituted into Eqn. (lognormlikelihood) to produce a likelihood equation in terms of <math>{{\sigma }_{{{T}'}}},</math> <math>t</math> and <math>R\ \ :</math>
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| ::<math>L({{\sigma }_{{{T}'}}},t/R)=\underset{i=1}{\overset{N}{\mathop \prod }}\,\frac{1}{{{x}_{i}}\cdot {{\sigma }_{{{T}'}}}\cdot \sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( \tfrac{\text{ln}({{x}_{i}})-\left( \text{ln}(t)-{{\sigma }_{{{T}'}}}\cdot {{\Phi }^{-1}}(1-R) \right)}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>
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| 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, Eqn. (lognormliketr) can be used to solve Eqn. (lratio3) for the values of interest.
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| '''Example 6:'''
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| {{Example: Lognormal Distribution Likelihood Ratio Bound (Time)}}
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| '''Example 7:'''
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| {{Example: Lognormal Distribution Likelihood Ratio Bound (Reliability)}}
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