1P-Exponential MLE Solution for Interval Data
 |
|
1P-Exponential MLE Solution for Interval Data |
This example validates the calculations for the MLE solution, likelihood ratio bound and Fisher Matrix bound for a 1-parameter exponential distribution with interval data in Weibull++ standard folios.
Reference Case
Example 7.1 on page 154 in the book Statistical Methods for Reliability Data by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998. The sample size of 200 data is used here.
Data
| Number in State | Last Inspected | State F/S | State End Time |
|---|---|---|---|
| 41 | 0 | F | 100 |
| 44 | 100 | F | 300 |
| 24 | 300 | F | 500 |
| 32 | 500 | F | 700 |
| 29 | 700 | F | 1000 |
| 21 | 1000 | F | 2000 |
| 9 | 2000 | F | 4000 |
Result
The cumulative distribution function for an exponential distribution is:
- [math]F(t)=1-e^{-\left( \frac{t}{\theta }\right )}\,\![/math]
The ML estimate [math]\hat{\theta}\,\![/math] = 572.3, and the standard deviation is [math]se_{\hat\theta}\,\![/math] = 41.72. Therefore the variance is 1740.56.
The 95% 2-sided confidence interval for [math]{\theta}\,\![/math] are:
- Based on the likelihood ratio, the confidence interval is [498, 662]. The calculation is based on
- [math]-2ln\left [ \frac{L(\theta)}{L(\hat{\theta})} \right ] = X^{2}_{(0.90,1)}\,\![/math]
- The two solutions of [math]\theta\,\![/math] in the above equation will be the confidence bounds for [math]\theta\,\![/math].
- Based on lognormal approximation, the confidence interval is [496, 660]. The calculation is:
- [math]\begin{alignat}{2} [\theta_{L},\theta_{U}]&= \hat{\theta}exp\left(\pm 1.96\times \frac{se_{\hat{\theta}}}{\hat{\theta}}\right)\\ &= \left[572.3\times exp \left(-1.96\times\frac{41.72}{572.3}\right),572.3\times exp \left(1.96\times\frac{41.72}{572.3}\right)\right]\\ &= [496,660]\\ \end{alignat}[/math]
Results in Weibull++
The ML estimator for [math]\theta\,\![/math] and its variance are 572.27 and 1740.52, respectively. They are given below.
The ML estimator for [math]\theta\,\![/math] and the variance are the same as the values given in the book.
The 95% 2-sided confidence interval for [math]\theta\,\![/math] are:
- Based on the likelihood ratio (Select LRB for the confidence bound), the confidence interval is:
- Based on lognormal approximation (select FM for the bound method), the confidence bounds are:
