Template:Example: Lognormal Distribution Likelihood Ratio Bound (Time)

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 $$R(t)=80%$$  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 $$R=0.80$$  and  $$\alpha =0.75$$  into the likelihood function, and varying  $$$$  until the maximum and minimum values of  $$t$$  are found. The following table gives the values of $$t$$  based on given values of  $$$$.

$$\begin{matrix} & {{t}_{1}} & {{t}_{2}} & & {{t}_{1}} & {{t}_{2}}  \\ 0.24 & 56.832 & 62.879 & 0.37 & 44.841 & 64.031 \\   0.25 & 54.660 & 64.287 & 0.38 & 44.494 & 63.454  \\   0.26 & 53.093 & 65.079 & 0.39 & 44.200 & 62.809  \\   0.27 & 51.811 & 65.576 & 0.40 & 43.963 & 62.093  \\   0.28 & 50.711 & 65.881 & 0.41 & 43.786 & 61.304  \\   0.29 & 49.743 & 66.041 & 0.42 & 43.674 & 60.436  \\   0.30 & 48.881 & 66.085 & 0.43 & 43.634 & 59.481  \\   0.31 & 48.106 & 66.028 & 0.44 & 43.681 & 58.426  \\   0.32 & 47.408 & 65.883 & 0.45 & 43.832 & 57.252  \\   0.33 & 46.777 & 65.657 & 0.46 & 44.124 & 55.924  \\   0.34 & 46.208 & 65.355 & 0.47 & 44.625 & 54.373  \\   0.35 & 45.697 & 64.983 & 0.48 & 45.517 & 52.418  \\   0.36 & 45.242 & 64.541 & {} & {} & {}  \\ \end{matrix}$$

This data set is represented graphically in the following contour plot:



As can be determined from the table, the lowest calculated value for $$t$$  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%.