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 $$\lambda $$, determine the 85% two-sided confidence bounds on the time estimate for a reliability of 90%. The ML estimate for the time at $$R(t)=90%$$ is $$\hat{t}=7.797.$$.

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 $$R=0.90$$ and $$\alpha =0.85$$ into the likelihood ratio bound equation. It now remains to find the values of $$t$$ which satisfy this equation. Since there is only one parameter, there are only two values of $$t$$ that will satisfy the equation. These values represent the $$\delta =85%$$ two-sided confidence limits of the time estimate $$\hat{t}$$. For our problem, the confidence limits are:


 * $${{\hat{t}}_{R=0.9}}=(4.359,16.033).$$