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===Likelihood Ratio Confidence Bounds===
#REDIRECT [[The_Lognormal_Distribution#Likelihood_Ratio_Confidence_Bounds]]
 
====Bounds on Parameters====
As covered in Chapter 5, 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 }_{{{T}'}}})=\underset{i=1}{\overset{N}{\mathop \prod }}\,f({{x}_{i}};{\mu }',{{\sigma }_{{{T}'}}})=\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}})-{\mu }'}{{{\sigma }_{{{T}'}}}} \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 }_{{{T}'}}}</math>  can be found which represent the maximum and minimum values that satisfy Eqn. (lratio3). 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 }_{{{T}'}}}} \right)</math>
 
This can be rearranged to the form:
 
::<math>{\mu }'=\text{ln}(t)-{{\sigma }_{{{T}'}}}\cdot {{\Phi }^{-1}}(1-R)</math>
 
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> 
 
::<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>
 
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.
 
====Example 6====
For the data given in Example 5, determine the two-sided 75% confidence bounds on the time estimate for a reliability of 80%.  The ML estimate for the time at  <math>R(t)=80%</math>  is 55.718.
=====Solution to Example 6=====
In this example, we are trying to determine the two-sided 75% confidence bounds on the time estimate of 55.718. This is accomplished by substituting  <math>R=0.80</math>  and  <math>\alpha =0.75</math>  into Eqn. (lognormliketr), and varying  <math>{{\sigma }_{{{T}'}}}</math>  until the maximum and minimum values of  <math>t</math>  are found. The following table gives the values of  <math>t</math>  based on given values of  <math>{{\sigma }_{{{T}'}}}</math> .
 
 
<center><math>\begin{matrix}
  {{\sigma }_{{{T}'}}} & {{t}_{1}} & {{t}_{2}} & {{\sigma }_{{{T}'}}} & {{t}_{1}} & {{t}_{2}}  \\
  0.24 & 56.832 & 62.879 & 0.37 & 44.841 & 64.031  \\
  0.25 & 54.660 & 64.287 & 0.38 & 44.494 & 63.454  \\
  0.26 & 53.093 & 65.079 & 0.39 & 44.200 & 62.809  \\
  0.27 & 51.811 & 65.576 & 0.40 & 43.963 & 62.093  \\
  0.28 & 50.711 & 65.881 & 0.41 & 43.786 & 61.304  \\
  0.29 & 49.743 & 66.041 & 0.42 & 43.674 & 60.436  \\
  0.30 & 48.881 & 66.085 & 0.43 & 43.634 & 59.481  \\
  0.31 & 48.106 & 66.028 & 0.44 & 43.681 & 58.426  \\
  0.32 & 47.408 & 65.883 & 0.45 & 43.832 & 57.252  \\
  0.33 & 46.777 & 65.657 & 0.46 & 44.124 & 55.924  \\
  0.34 & 46.208 & 65.355 & 0.47 & 44.625 & 54.373  \\
  0.35 & 45.697 & 64.983 & 0.48 & 45.517 & 52.418  \\
  0.36 & 45.242 & 64.541 & {} & {} & {}  \\
\end{matrix}</math></center>
 
 
This data set is represented graphically in the following contour plot:
 
[[Image:ldachp9ex6.gif|thumb|center|400px| ]]  
 
As can be determined from the table, the lowest calculated value for  <math>t</math>  is 43.634, while the highest is 66.085. These represent the two-sided 75% confidence limits on the time at which reliability is equal to 80%.
 
====Example 7====
For the data given in Example 5, determine the two-sided 75% confidence bounds on the reliability estimate for  <math>t=65</math> .  The ML estimate for the reliability at  <math>t=65</math>  is 64.261%.
=====Solution to Example 7=====
In this example, we are trying to determine the two-sided 75% confidence bounds on the reliability estimate of 64.261%. This is accomplished by substituting  <math>t=65</math>  and  <math>\alpha =0.75</math>  into Eqn. (lognormliketr), and varying  <math>{{\sigma }_{{{T}'}}}</math>  until the maximum and minimum values of  <math>R</math>  are found. The following table gives the values of  <math>R</math>  based on given values of  <math>{{\sigma }_{{{T}'}}}</math> .
 
 
<center><math>\begin{matrix}
  {{\sigma }_{{{T}'}}} & {{R}_{1}} & {{R}_{2}} & {{\sigma }_{{{T}'}}} & {{R}_{1}} & {{R}_{2}}  \\
  0.24 & 61.107% & 75.910% & 0.37 & 43.573% & 78.845%  \\
  0.25 & 55.906% & 78.742% & 0.38 & 43.807% & 78.180%  \\
  0.26 & 55.528% & 80.131% & 0.39 & 44.147% & 77.448%  \\
  0.27 & 50.067% & 80.903% & 0.40 & 44.593% & 76.646%  \\
  0.28 & 48.206% & 81.319% & 0.41 & 45.146% & 75.767%  \\
  0.29 & 46.779% & 81.499% & 0.42 & 45.813% & 74.802%  \\
  0.30 & 45.685% & 81.508% & 0.43 & 46.604% & 73.737%  \\
  0.31 & 44.857% & 81.387% & 0.44 & 47.538% & 72.551%  \\
  0.32 & 44.250% & 81.159% & 0.45 & 48.645% & 71.212%  \\
  0.33 & 43.827% & 80.842% & 0.46 & 49.980% & 69.661%  \\
  0.34 & 43.565% & 80.446% & 0.47 & 51.652% & 67.789%  \\
  0.35 & 43.444% & 79.979% & 0.48 & 53.956% & 65.299%  \\
  0.36 & 43.450% & 79.444% & {} & {} & {}  \\
\end{matrix}</math></center>
 
 
This data set is represented graphically in the following contour plot:
 
[[Image:ldachp9ex7.gif|thumb|center|400px| ]]
 
As can be determined from the table, the lowest calculated value for  <math>R</math>  is 43.444%, while the highest is 81.508%. These represent the two-sided 75% confidence limits on the reliability at  <math>t=65</math> .

Latest revision as of 06:06, 13 August 2012

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