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1P-Exponential MLE Solution with Right Censored Data
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This example validates the calculations for the MLE solution and Fisher Matrix bound for a 1-parameter exponential distribution with right censored and complete failure data in Weibull++ standard folios.
Reference Case
The formulas on page 166 in the book Statistical Methods for Reliability Data by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998.
- [math]\displaystyle{ \hat{\theta}=\frac{TTT}{r}\ \ and\ \ se_{\hat{\theta}} = \frac{\hat{\theta}}{\sqrt{r}}\,\! }[/math]
where TTT is the total test time and r is the number of failures.
Data
| Number in State
|
State F or S
|
Time to Failure
|
| 1 |
F |
16
|
| 1 |
F |
34
|
| 1 |
F |
53
|
| 1 |
F |
75
|
| 1 |
F |
93
|
| 1 |
F |
120
|
| 4 |
S |
200
|
Result
- [math]\displaystyle{ \begin{align}
\hat{\theta} =& \frac{TTT}{r} = \frac{16+34+53+75+93+120+4\times 200}{6} = \frac{1191}{6} = 198.5 \\
\\
se_{\hat{\theta}} =& \frac{\theta}{\sqrt{r}} = \frac{198.5}{\sqrt{6}} = 81.037 \\
\end{align}\,\! }[/math]
So the variance of [math]\displaystyle{ \hat{\theta}\,\! }[/math] is 6567.04
Results in Weibull++
