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| ==Simple Actuarial Method==
| | #REDIRECT [[Non-Parametric Life Data Analysis]] |
| The simple actuarial method is an easy-to-use form of nonparametric data analysis that can be used for multiply censored data that are arranged in intervals. This method is based on calculating the number of failures in a time interval, <math>{{r}_{j}},</math> versus the number of operating units in that time period, <math>{{n}_{j}}</math> . The equation for the reliability estimator for the standard actuarial method is given by:
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| ::<math>\widehat{R}({{t}_{i}})=\underset{j=1}{\overset{i}{\mathop \prod }}\,\left( 1-\frac{{{r}_{j}}}{{{n}_{j}}} \right),\text{ }i=1,...,m</math>
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| :where:
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| ::<math>\begin{align}
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| & m= & \text{the total number of intervals} \\
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| & n= & \text{the total number of units}
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| \end{align}</math>
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| The variable <math>{{n}_{i}}</math> is defined by:
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| ::<math>{{n}_{i}}=n-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{s}_{j}}-\underset{j=0}{\overset{i-1}{\mathop \sum }}\,{{r}_{j,}}\text{ }i=1,...,m</math>
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| :where:
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| ::<math>\begin{align}
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| & {{r}_{j}}= & \text{the number of failures in interval }j \\
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| & {{s}_{j}}= & \text{the number of suspensions in interval }j
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| \end{align}</math>
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| ====Example 10====
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| A group of 55 units are put on a life test during which the units are evaluated every 50 hours, with the following results:
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| <center><math>\begin{matrix}
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| Start & End & Number of & Number of \\
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| Time & Time & Failures, {{r}_{i}} & Suspensions, {{s}_{i}} \\
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| 0 & 50 & 2 & 4 \\
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| 50 & 100 & 0 & 5 \\
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| 100 & 150 & 2 & 2 \\
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| 150 & 200 & 3 & 5 \\
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| 200 & 250 & 2 & 1 \\
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| 250 & 300 & 1 & 2 \\
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| 300 & 350 & 2 & 1 \\
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| 350 & 400 & 3 & 3 \\
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| 400 & 450 & 3 & 4 \\
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| 450 & 500 & 1 & 2 \\
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| 500 & 550 & 2 & 1 \\
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| 550 & 600 & 1 & 0 \\
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| 600 & 650 & 2 & 1 \\
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| \end{matrix}</math></center>
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| =====Solution to Example 10=====
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| 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):
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| <center><math>\begin{matrix}
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| Start & End & Number of & Number of & Available & {} & {} \\
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| Time & Time & Failures, {{r}_{i}} & Suspensions, {{s}_{i}} & Units, {{n}_{i}} & 1-\tfrac{{{r}_{j}}}{{{n}_{j}}} & \mathop{}_{}^{}1-\tfrac{{{r}_{j}}}{{{n}_{j}}} \\
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| 0 & 50 & 2 & 4 & 55 & 0.964 & 0.964 \\
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| 50 & 100 & 0 & 5 & 49 & 1.000 & 0.964 \\
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| 100 & 150 & 2 & 2 & 44 & 0.955 & 0.920 \\
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| 150 & 200 & 3 & 5 & 40 & 0.925 & 0.851 \\
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| 200 & 250 & 2 & 1 & 32 & 0.938 & 0.798 \\
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| 250 & 300 & 1 & 2 & 29 & 0.966 & 0.770 \\
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| 300 & 350 & 2 & 1 & 26 & 0.923 & 0.711 \\
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| 350 & 400 & 3 & 3 & 23 & 0.870 & 0.618 \\
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| 400 & 450 & 3 & 4 & 17 & 0.824 & 0.509 \\
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| 450 & 500 & 1 & 2 & 10 & 0.900 & 0.458 \\
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| 500 & 550 & 2 & 1 & 7 & 0.714 & 0.327 \\
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| 550 & 600 & 1 & 0 & 4 & 0.750 & 0.245 \\
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| 600 & 650 & 2 & 1 & 3 & 0.333 & 0.082 \\
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| \end{matrix}</math></center>
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| As can be determined from the preceding table, the reliability estimates for the failure times are:
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| <center><math>\begin{matrix}
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| Failure Period & Reliability \\
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| End Time & Estimate \\
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| 50 & 96.4% \\
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| 150 & 92.0% \\
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| 200 & 85.1% \\
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| 250 & 79.8% \\
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| 300 & 77.0% \\
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| 350 & 71.1% \\
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| 400 & 61.8% \\
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| 450 & 50.9% \\
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| 500 & 45.8% \\
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| 550 & 32.7% \\
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| 600 & 24.5% \\
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| 650 & 8.2% \\
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| \end{matrix}</math></center>
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