Lognormal MLE Solution with Right Censored Data

This example validates the calculations for the Lognormal MLE solution with Fisher matrix bound for right censored data in Weibull++ standard folios.

The data set on page 199 of the book Statistical Methods for Reliability Data by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998 is used.


 * The MLE solution is $$\hat{\mu} = 6.56, \ \hat{\sigma} = 0.543\,\!$$.


 * The variance and covariance matrix is

\sum =\begin{bmatrix} 0.0581 & 0.0374 \\ 0.0374 & 0.0405 \end{bmatrix}\,\!$$


 * The following picture shows the MLE solution and the variance/covariance matrix:




 * The Fisher matrix bound for parameters are:


 * For Ln-mu (using normal approximation of Eqn. 8.7 on page 187):


 * $$[\hat{\mu}'_{L}, \hat{\mu}'_{U}] = \hat{\mu}' \pm z_{(1-\alpha / 2)}se_{\hat{\mu}'}\,\!$$


 * For a confidence level of 0.95, it is:


 * $$\begin{alignat}{2}

[\hat{\mu}'_{L}, \hat{\mu}'_{U}] =& \hat{\mu}' \pm z_{(1-\alpha /2)}se_{\hat{\mu}'}\\ =& 6.564256 \pm 1.96 \times (0.0581)^{0.5}\\ =& [6.0918, 7.0366933]\\ \end{alignat}\,\!$$


 * For Ln-Std (using log-normal approximation of Eqn. 8.8 on page 188):


 * $$[\hat{\sigma}'_{L}, \hat{\sigma}'_{U}] = \hat{\sigma}'exp (\pm \frac{z_{(1-\alpha /2)}se_{\hat{\sigma}'}}{\hat{\sigma}'})\,\!$$


 * For confidence level of 0.95, it is:


 * $$\begin{alignat}{2}

[\hat{\sigma}'_{L}, \hat{\sigma}'_{U}] =& \hat{\sigma}'exp (\pm \frac{z_{(1-\alpha /2)}se_{\hat{\sigma}'}}{\hat{\sigma}'})\\ =& 0.534049 \times exp(\pm \tfrac{1.96\times 0.040562^{0.5}}{0.534049})\\ =& [0.255, 1.118]\\ \end{alignat} \,\!$$


 * The following picture shows the results in Weibull++: