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

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'''Lognormal Distribution Likelihood Ratio Bound Example (Time)'''
#REDIRECT [[The Lognormal Distribution]]
 
For the data given in [[Lognormal Example 5 Data|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'''
 
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 the likelihood function, and varying  <math>{{\sigma' }}</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' }}</math> .
 
<center><math>\begin{matrix}
  {{\sigma' }} & {{t}_{1}} & {{t}_{2}} & {{\sigma' }} & {{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:WB.10 time vs sigma.png|center|350px| ]]  
 
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%.

Latest revision as of 03:38, 13 August 2012

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