Template:Example: Lognormal Distribution Likelihood Ratio Bound (Time): Difference between revisions
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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. | 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''' | '''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 | 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} | <center><math>\begin{matrix} | ||
{{\sigma | {{\sigma' }} & {{t}_{1}} & {{t}_{2}} & {{\sigma' }} & {{t}_{1}} & {{t}_{2}} \\ | ||
0.24 & 56.832 & 62.879 & 0.37 & 44.841 & 64.031 \\ | 0.24 & 56.832 & 62.879 & 0.37 & 44.841 & 64.031 \\ | ||
0.25 & 54.660 & 64.287 & 0.38 & 44.494 & 63.454 \\ | 0.25 & 54.660 & 64.287 & 0.38 & 44.494 & 63.454 \\ | ||
Revision as of 23:25, 13 February 2012
Lognormal Distribution Likelihood Ratio Bound Example (Time)
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]\displaystyle{ 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]\displaystyle{ R=0.80 }[/math] and [math]\displaystyle{ \alpha =0.75 }[/math] into the likelihood function, and varying [math]\displaystyle{ {{\sigma' }} }[/math] until the maximum and minimum values of [math]\displaystyle{ t }[/math] are found. The following table gives the values of [math]\displaystyle{ t }[/math] based on given values of [math]\displaystyle{ {{\sigma' }} }[/math] .
This data set is represented graphically in the following contour plot:
As can be determined from the table, the lowest calculated value for [math]\displaystyle{ 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%.