1P-Exponential MLE Solution with Right Censored Data: Difference between revisions

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{{Reference Example}}
{{Reference Example}}
Compare the MLE solution and Fisher Matrix bound for a 1-parameter exponential distribution with right censored and complete failure data.   
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.   




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{{Reference_Example_Heading3}}
{{Reference_Example_Heading3}}


::<math>\hat{\theta} = \frac{TTT}{r} = \frac{16+34+53+75+93+120+4\times 200}{6} = \frac{1191}{6} = 198.5\,\!</math>
::<math>\begin{align}
 
\hat{\theta} =& \frac{TTT}{r} = \frac{16+34+53+75+93+120+4\times 200}{6} = \frac{1191}{6} = 198.5 \\
 
\\
::<math>se_{\hat{\theta}} = \frac{\theta}{\sqrt{6}} = \frac{198.5}{\sqrt{6}} = 81.037 \,\!</math>
se_{\hat{\theta}} =& \frac{\theta}{\sqrt{r}} = \frac{198.5}{\sqrt{6}} = 81.037 \\
\end{align}\,\!</math>




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1P-Exponential MLE Solution with Right Censored Data

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++


1PE rcensored data.png
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