1P-Exponential MLE Solution for Interval Data

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 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:

$F(t)=1-e^{-\left( \frac{t}{\theta }\right )}\,\!$

The ML estimate $\hat{\theta}\,\!$ = 572.3, and the standard deviation is $se_{\hat\theta}\,\!$ = 41.72. Therefore the variance is 1740.56.

The 95% 2-sided confidence interval for ${\theta}\,\!$ are:

• Based on the likelihood ratio, the confidence interval is [498, 662]. The calculation is based on
$-2ln\left [ \frac{L(\theta)}{L(\hat{\theta})} \right ] = X^{2}_{(0.90,1)}\,\!$

The two solutions of $\theta\,\!$ in the above equation will be the confidence bounds for $\theta\,\!$.

• Based on lognormal approximation, the confidence interval is [496, 660]. The calculation is:
\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}

Results in Weibull++

The ML estimator for $\theta\,\!$ and its variance are 572.27 and 1740.52, respectively. They are given below.

The ML estimator for $\theta\,\!$ and the variance are the same as the values given in the book.

The 95% 2-sided confidence interval for $\theta\,\!$ 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: