Lognormal MLE Solution with Right Censored Data

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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.


Reference Case

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.


Data

Number in State State F or S Time to Failure
1 F 252
1 F 315
1 F 369
1 F 403
1 F 474
15 S 500


Result

  • The MLE solution is [math]\hat{\mu} = 6.56, \ \hat{\sigma} = 0.543\,\![/math].
  • The variance and covariance matrix is
[math] \sum =\begin{bmatrix} 0.0581 & 0.0374 \\ 0.0374 & 0.0405 \end{bmatrix}\,\![/math]


Results in Weibull++


  • The following picture shows the MLE solution and the variance/covariance matrix:
Lg right censored matrix.png


  • The Fisher matrix bound for parameters are:
For Ln-mu (using normal approximation of Eqn. 8.7 on page 187):
[math][\hat{\mu}'_{L}, \hat{\mu}'_{U}] = \hat{\mu}' \pm z_{(1-\alpha / 2)}se_{\hat{\mu}'}\,\![/math]


For a confidence level of 0.95, it is:
[math]\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}\,\![/math]


For Ln-Std (using log-normal approximation of Eqn. 8.8 on page 188):
[math][\hat{\sigma}'_{L}, \hat{\sigma}'_{U}] = \hat{\sigma}'exp (\pm \frac{z_{(1-\alpha /2)}se_{\hat{\sigma}'}}{\hat{\sigma}'})\,\![/math]


For confidence level of 0.95, it is:
[math]\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} \,\![/math]


The following picture shows the results in Weibull++:
Lg right censored bounds.png