Template:Test Design Using Expected Failure Time Plots

One of the new features in Weibull++ 8 is the Expected Failure Time Calculation (EFTC) tool. Given the population size and the failure time distribution $$F(t) $$, the EFTC tool estimates the expected value of the nth failure and its confidence interval at any confidence level.

The EFTC tool works by first calculating the probability of failure F as a function of the sample size n and number of failures r. This function is given by the equation $$ f\left(n,r,CL\right)=\frac{1}{1+\frac{n-r+1}{rF_{2r,2\left(n-r+1\right),1-CL}}} $$ where CL is the confidence level.

For example, given n = 4, r = 2 and CL = 0.5, the median rank for F is $$ f\left(4,2,0.5\right)=\frac{1}{1+\frac{4-2+1}{2F_{4,2\left(3\right),0.5}}}=0.385728 $$ The 80% 2-sided confidence interval on the probability of failure F is bounded by $$ f\left(4,2,0.1\right)=\frac{1}{1+\frac{4-2+1}{2F_{4,2\left(3\right),0.9}}}=0.142559 $$ and $$ f\left(4,2,0.9\right)=\frac{1}{1+\frac{4-2+1}{2F_{4,2\left(3\right),0.1}}}=0.679539 $$ With the probabilities of failure known, and assuming the failure distribution is exponentially distributed with parameter \(\lambda\,\!\) = 1, the expected times to failure can be calculated.

For example, the median time to the second failure is calculated as $$ t=-\frac{ln\left(1-F\right)}{\lambda}=-\frac{1-0.385728}{1}=0.48732 $$ Similarly, the 10th percentile and 90th percentiles of time to failure are calculated as $$ t=-\frac{1-0.142559}{1}=0.153803 $$ and $$ t=-\frac{1-0.679539}{1}=1.137995 $$ The above results can also be seen in Weibull++ as shown below.