Template:Non-Parametric LDA Example (Simple Actuarial Method)

Non-Parametric LDA Example (Simple Actuarial Method)

Problem Statement (Simple Actuarial Method)

A group of 55 units are put on a life test during which the units are evaluated every 50 hours, with the following results:

[math]\displaystyle{ \begin{matrix} Start & End & Number of & Number of \\ Time & Time & Failures, {{r}_{i}} & Suspensions, {{s}_{i}} \\ 0 & 50 & 2 & 4 \\ 50 & 100 & 0 & 5 \\ 100 & 150 & 2 & 2 \\ 150 & 200 & 3 & 5 \\ 200 & 250 & 2 & 1 \\ 250 & 300 & 1 & 2 \\ 300 & 350 & 2 & 1 \\ 350 & 400 & 3 & 3 \\ 400 & 450 & 3 & 4 \\ 450 & 500 & 1 & 2 \\ 500 & 550 & 2 & 1 \\ 550 & 600 & 1 & 0 \\ 600 & 650 & 2 & 1 \\ \end{matrix} }[/math]

Solution (Simple Actuarial Method)

The reliability estimates for the simple actuarial method can be obtained by expanding the data table to include terms used in calculation of the reliability estimates for Eqn. (simpact):

[math]\displaystyle{ \begin{matrix} Start & End & Number of & Number of & Available & {} & {} \\ Time & Time & Failures, {{r}_{i}} & Suspensions, {{s}_{i}} & Units, {{n}_{i}} & 1-\tfrac{{{r}_{j}}}{{{n}_{j}}} & \mathop{}_{}^{}1-\tfrac{{{r}_{j}}}{{{n}_{j}}} \\ 0 & 50 & 2 & 4 & 55 & 0.964 & 0.964 \\ 50 & 100 & 0 & 5 & 49 & 1.000 & 0.964 \\ 100 & 150 & 2 & 2 & 44 & 0.955 & 0.920 \\ 150 & 200 & 3 & 5 & 40 & 0.925 & 0.851 \\ 200 & 250 & 2 & 1 & 32 & 0.938 & 0.798 \\ 250 & 300 & 1 & 2 & 29 & 0.966 & 0.770 \\ 300 & 350 & 2 & 1 & 26 & 0.923 & 0.711 \\ 350 & 400 & 3 & 3 & 23 & 0.870 & 0.618 \\ 400 & 450 & 3 & 4 & 17 & 0.824 & 0.509 \\ 450 & 500 & 1 & 2 & 10 & 0.900 & 0.458 \\ 500 & 550 & 2 & 1 & 7 & 0.714 & 0.327 \\ 550 & 600 & 1 & 0 & 4 & 0.750 & 0.245 \\ 600 & 650 & 2 & 1 & 3 & 0.333 & 0.082 \\ \end{matrix} }[/math]

As can be determined from the preceding table, the reliability estimates for the failure times are:

[math]\displaystyle{ \begin{matrix} Failure Period & Reliability \\ End Time & Estimate \\ 50 & 96.4% \\ 150 & 92.0% \\ 200 & 85.1% \\ 250 & 79.8% \\ 300 & 77.0% \\ 350 & 71.1% \\ 400 & 61.8% \\ 450 & 50.9% \\ 500 & 45.8% \\ 550 & 32.7% \\ 600 & 24.5% \\ 650 & 8.2% \\ \end{matrix} }[/math]