Template:Example: Lognormal Distribution Likelihood Ratio Bound (Reliability): Difference between revisions

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'''Lognormal Distribution Likelihood Ratio Bound Example (Reliability)'''
#REDIRECT [[The Lognormal Distribution]]
 
For the data given in [[Lognormal Example 5 Data|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'''
 
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 the likelihood function, and varying  <math>{{\sigma'}}</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' }}</math> .
 
 
<center><math>\begin{matrix}
  {{\sigma'}} & {{R}_{1}} & {{R}_{2}} & {{\sigma'}} & {{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:WB.10 reliability v sigma.png|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 03:43, 13 August 2012

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