Template:Exponential Distribution Example: Likelihood Ratio Bound for Reliability: Difference between revisions
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'''Likelihood Ratio Bound on Reliability''' | |||
For the data given in Example 5, determine the 85% two-sided confidence bounds on the reliability estimate for a <math>t=50</math>. The ML estimate for the time at <math>t=50</math> is <math>\hat{R}=50.881%</math>. | For the data given in Example 5, determine the 85% two-sided confidence bounds on the reliability estimate for a <math>t=50</math>. The ML estimate for the time at <math>t=50</math> is <math>\hat{R}=50.881%</math>. | ||
'''Solution | '''Solution''' | ||
In this example, we are trying to determine the 85% two-sided confidence bounds on the reliability estimate of 50.881%. This is accomplished by substituting <math>t=50</math> and <math>\alpha =0.85</math> into the likelihood ratio bound equation. It now remains to find the values of <math>R</math> which satisfy this equation. Since there is only one parameter, there are only two values of <math>t</math> that will satisfy the equation. These values represent the <math>\delta =85%</math> two-sided confidence limits of the reliability estimate <math>\hat{R}</math>. For our problem, the confidence limits are: | In this example, we are trying to determine the 85% two-sided confidence bounds on the reliability estimate of 50.881%. This is accomplished by substituting <math>t=50</math> and <math>\alpha =0.85</math> into the likelihood ratio bound equation. It now remains to find the values of <math>R</math> which satisfy this equation. Since there is only one parameter, there are only two values of <math>t</math> that will satisfy the equation. These values represent the <math>\delta =85%</math> two-sided confidence limits of the reliability estimate <math>\hat{R}</math>. For our problem, the confidence limits are: | ||
::<math>{{\hat{R}}_{t=50}}=(29.861%,71.794%)</math> | ::<math>{{\hat{R}}_{t=50}}=(29.861%,71.794%)</math> | ||
Revision as of 18:29, 9 February 2012
Likelihood Ratio Bound on Reliability
For the data given in Example 5, determine the 85% two-sided confidence bounds on the reliability estimate for a [math]\displaystyle{ t=50 }[/math]. The ML estimate for the time at [math]\displaystyle{ t=50 }[/math] is [math]\displaystyle{ \hat{R}=50.881% }[/math].
Solution
In this example, we are trying to determine the 85% two-sided confidence bounds on the reliability estimate of 50.881%. This is accomplished by substituting [math]\displaystyle{ t=50 }[/math] and [math]\displaystyle{ \alpha =0.85 }[/math] into the likelihood ratio bound equation. It now remains to find the values of [math]\displaystyle{ R }[/math] which satisfy this equation. Since there is only one parameter, there are only two values of [math]\displaystyle{ t }[/math] that will satisfy the equation. These values represent the [math]\displaystyle{ \delta =85% }[/math] two-sided confidence limits of the reliability estimate [math]\displaystyle{ \hat{R} }[/math]. For our problem, the confidence limits are:
- [math]\displaystyle{ {{\hat{R}}_{t=50}}=(29.861%,71.794%) }[/math]