# Topp-Leone Modified Weibull Model: Theory and Applications to Medical and Engineering Data
## 6 Applications
### COVID-19 Data  
The data set given as follows: 1, 1, 2, 4, 5, 1, 1, 3, 6, 6, 4, 1, 5, 6, 6, 8, 5, 7, 7, 9, 9, 15, 17, 11, 13, 5, 14, 5, 13, 9, 19, 15, 11, 14, 12, 11, 7, 13, 10, 20, 22, 21, 12, 14, 9, 14, 7, 16, 17, 13, 21, 11, 8, 11, 12, 15, 21, 20, 18, 15, 14, 21, 16, 11, 28, 29, 19, 14, 19, 29, 34, 34, 46, 47, 36.00, 38, 40, 32, 39, 34, 35, 36, 35, 45, 62. The most recent data, used by [4], revealed the daily fatality confirmed cases attributable to COVID-19. The data consists of 89 observed values, with 18.72 reported deaths on average every day.  
By using equations Equations (22)-(25) to obtain the MLEs of the model parameters with standard error (SE) and the results are reported in Table 7. The estimated density with histogram of waiting times data for \(TLMW\) distribution and other distributions are displayed in Figure 5. The estimated cumulative and the fitted \(TLMW\) distribution and other distributions for the waiting times data set are displayed in Figure 6. The estimated PP and QQ plot for \(TLMW\) distribution are displayed for waiting times data in Figure 7.  
\begin{table}
\begin{tabular}{l l l l l l l l l l l} \hline \multicolumn{2}{c}{**Third Combination**} & \multicolumn{3}{c}{**ML**} & \multicolumn{3}{c}{**MPS**} & \multicolumn{3}{c}{**Bayesian**} \\ \hline \(\beta\) & n & & **RB** & **MSEr** & **LACI** & **RB** & **MSEr** & **LACI** & **RB** & **MSEr** & **LCCI** \\ \hline \multirow{4}{*}{30} & & \(\alpha\) & 1.0510 & 3.1518 & 4.9250 & 1.4816 & 1.4329 & 3.6877 & 0.4612 & 0.3538 & 1.8668 \\  & & \(\beta\) & 0.5033 & 3.9465 & 5.0629 & 0.3231 & 1.5708 & 3.1161 & 0.5670 & 1.4974 & 3.8887 \\  & & \(\theta\) & 0.3385 & 0.4865 & 2.5479 & 0.1994 & 0.0549 & 0.7076 & 0.2919 & 0.1676 & 2.2720 \\  & & \(\lambda\) & 0.7493 & 5.0224 & 6.5419 & 1.1926 & 6.5029 & 3.5369 & 0.7382 & 4.8340 & 5.9546 \\ \cline{2-11}  & & \(\alpha\) & 0.9208 & 1.3430 & 2.5436 & 1.3559 & 0.9532 & 2.3051 & 0.2860 & 0.2647 & 1.6196 \\  & & \(\beta\) & 0.3678 & 1.5569 & 2.2854 & 0.3032 & 1.0617 & 1.8992 & 0.3258 & 1.2887 & 2.1018 \\  & & \(\theta\) & 0.1766 & 0.2832 & 0.4069 & 0.1677 & 0.0262 & 0.4000 & 0.1607 & 0.1545 & 0.3666 \\  & & \(\lambda\) & 0.6748 & 3.5248 & 7.1293 & 0.9328 & 4.5597 & 2.8017 & 0.6817 & 2.6870 & 4.4917 \\ \hline \multirow{4}{*}{150} & & \(\alpha\) & 0.8160 & 1.3064 & 2.7226 & 0.9961 & 0.8072 & 1.5698 & 0.2892 & 0.2370 & 1.5071 \\  & & \(\beta\) & 0.3576 & 1.2649 & 2.7693 & 0.2941 & 0.8905 & 1.3131 & 0.3148 & 0.8488 & 2.0209 \\ \cline{1-1}  & & \(\theta\) & 0.1575 & 0.1372 & 0.3922 & 0.1571 & 0.0226 & 0.3097 & 0.1332 & 0.1354 & 0.2539 \\ \cline{1-1}  & & \(\lambda\) & 0.4965 & 1.2215 & 4.9840 & 0.9337 & 2.5183 & 2.3962 & 0.4603 & 1.0261 & 3.0571 \\ \hline \end{tabular}
\end{table}
Table 6: _Cont._  
The 95% CIs for the parameters \(\alpha\), \(\beta\), \(\theta\) and \(\lambda\) are presented for MLE and Bayesian estimation methods in Table 9. Figure 8 displayed contour plot of parameters for \(TLMW\) model to check uniqueness and max-value of likelihood for COVID-19 data. While, Figure 9 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 COVID-19 data. Figure 10 discussed Auto-correlation Function (ACF) test. Auto-  
Figure 8: Contour plot for parameters of \(TLMW\) model for COVID-19 data.  
Figure 9: MCMC plots for parameters of \(TLMW\) model for COVID-19 data.