# Topp-Leone Modified Weibull Model: Theory and Applications to Medical and Engineering Data
## 6 Applications
### Tensile Strength for Single Carbon Fibres (in GPa)  
The data set given in Table 18 contains the tensile strength for single carbon fibres (in GPa). The data set finds its source in [55]. We use equations Equations (22)-(25) to obtain the MLEs of the model parameters and the results are reported in Table 19. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Models** & & \(\mathbf{\alpha}\) & \(\mathbf{\beta}\) & \(\mathbf{\theta}\) & \(\mathbf{\lambda}\) & \(\mathbf{\delta}\) \\ \hline \multirow{2}{*}{\(TLMW\)} & estimates & 0.2145 & 0.1057 & 3.2053 & 2.2749 & \\  & SE & 0.1081 & 0.0346 & 0.7106 & 1.1533 & \\ \hline \multirow{2}{*}{OLLMW} & estimates & 28.7708 & 0.0560 & 0.6093 & 0.0125 & \\  & SE & 39.8918 & 0.0814 & 0.1193 & 0.0310 & \\ \hline \multirow{2}{*}{KW} & estimates & 1.3752 & 0.1402 & 1.2141 & 3.0059 & \\  & SE & 0.0409 & 0.0194 & 0.0306 & 0.0194 & \\ \hline \multirow{2}{*}{GMW} & estimates & 0.4864 & 4.4323 & 0.8582 & 0.3467 & \\  & SE & 0.7405 & 8.6912 & 0.2825 & 1.0545 & \\ \hline \multirow{2}{*}{EOWL} & estimates & 2.6361 & 0.1152 & 8.7047 & 19.2595 & \\  & SE & 0.4997 & 0.2885 & 4.9097 & 10.6442 & \\ \hline \multirow{2}{*}{WL} & estimates & 0.1399 & 2.5597 & 2.0355 & 2.1477 & \\  & SE & 4.5871 & 2.4273 & 6.7471 & 31.2428 & \\ \hline \multirow{2}{*}{MAOPEW} & estimates & 0.8564 & 3.0230 & 0.0117 & 1.5489 & 0.0055 \\  & SE & 1.3615 & 0.8091 & 0.0091 & 0.3239 & 0.0036 \\ \hline \multirow{2}{*}{EGAPEx} & estimates & 141.1326 & 2.4787 & 17.2979 & 0.4888 & \\  & SE & 52.8014 & 0.9138 & 12.7006 & 0.1625 & \\ \hline \hline \end{tabular}
\end{table} 
Table 19: MLE for parameters of \(TLMW\) and comparative models for single carbon fibers data. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
0.312 & 0.944 & 1.063 & 1.272 & 1.434 & 1.566 & 1.697 & 1.848 & 2.128 \\
0.314 & 0.958 & 1.098 & 1.274 & 1.435 & 1.57 & 1.726 & 1.88 & 2.233 \\
0.479 & 0.966 & 1.14 & 1.301 & 1.478 &The estimated density with histogram of single carbon fibers data for \(TLMW\) distribution and other distributions are displayed in Figure 23. The estimated cumulative and the fitted \(TLMW\) distribution and other distributions for the second set are displayed in Figure 24. The estimated PP and QQ plot for \(TLMW\) distribution are displayed for single carbon fibers data in Figure 25.  
Now, applying the formal goodness-of-fit tests in order to verify which distribution fits better to single carbon fibers data set. The measures AKIC, BIC, CAKIC, HQIC and the KSD with PVKS (See Table 20) are considered. The 95% CIs for the parameters \(\alpha\), \(\beta\), \(\theta\) and \(\lambda\) are presented for MLE and Bayesian estimation methods in Table 21. Figure 26 displayed contour plot of parameters for \(TLMW\) model to check uniqueness and max-value of likelihood for single carbon fibers data. While, Figure 27 confirmed convergence of MCMC results, which show the MCMC samples as a pairs plot, with the trace plot, and marginal frequency of proposed distribution as normal distribution for each parameter on the diagonal for parameters of \(TLMW\) model for single carbon fibers. Figure 28 discussed ACF test. Auto-correlation for MCMC is the correlation between a iteration series with a lagged version of itself. The correlation between the iteration series and itself, which is where the ACF starts with a lag of 0, yields a correlation of 1.