Template:Example: Normal Distribution Likelihood Ratio Bound (Reliability)

Normal Distribution Likelihood Ratio Bound Example (Reliability)

For the data given in Example 5, determine the two-sided 80% confidence bounds on the reliability estimate for [math]\displaystyle{ t=30 }[/math] . The ML estimate for the reliability at [math]\displaystyle{ t=30 }[/math] is 45.739%.

Solution In this example, we are trying to determine the two-sided 80% confidence bounds on the reliability estimate of 45.739%. This is accomplished by substituting [math]\displaystyle{ t=30 }[/math] and [math]\displaystyle{ \alpha =0.8 }[/math] into the likelihood ratio equation for normal distribution, and varying [math]\displaystyle{ \sigma }[/math] until the maximum and minimum values of [math]\displaystyle{ R }[/math] are found. The following table gives the values of [math]\displaystyle{ R }[/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{ R }[/math] is 24.776%, while the highest is 68.000%. These represent the 80% two-sided confidence limits on the reliability at [math]\displaystyle{ t=30 }[/math]