Template:Exponential Distribution Example: Likelihood Ratio Bound for Time

Example 6: Likelihood Ratio Bound for Time

For the data given in Example 5: Likelihood Ratio Bound for [math]\displaystyle{ \lambda }[/math], determine the 85% two-sided confidence bounds on the time estimate for a reliability of 90%. The ML estimate for the time at [math]\displaystyle{ R(t)=90% }[/math] is [math]\displaystyle{ \hat{t}=7.797. }[/math].

Solution to Example 6

In this example, we are trying to determine the 85% two-sided confidence bounds on the time estimate of 7.797. This is accomplished by substituting [math]\displaystyle{ R=0.90 }[/math] and [math]\displaystyle{ \alpha =0.85 }[/math] into the likelihood ratio bound equation. It now remains to find the values of [math]\displaystyle{ t }[/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 time estimate [math]\displaystyle{ \hat{t} }[/math]. For our problem, the confidence limits are:

[math]\displaystyle{ {{\hat{t}}_{R=0.9}}=(4.359,16.033). }[/math]