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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:
 
::<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>
 
:where:
 
::<math>\begin{align}
  & m= & \text{the total number of intervals} \\
& n= & \text{the total number of units} 
\end{align}</math>
 
The variable  <math>{{n}_{i}}</math>  is defined by:
 
::<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>
 
:where:
 
::<math>\begin{align}
  & {{r}_{j}}= & \text{the number of failures in interval }j \\
& {{s}_{j}}= & \text{the number of suspensions in interval }j 
\end{align}</math>
 
 
====Example 10====
A group of 55 units are put on a life test during which the units are evaluated every 50 hours, with the following results:
 
<center><math>\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></center>
 
 
 
=====Solution to Example 10=====
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):
 
<center><math>\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></center>
 
As can be determined from the preceding table, the reliability estimates for the failure times are:
 
<center><math>\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></center>

Latest revision as of 07:35, 29 June 2012

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