# Topp-Leone Modified Weibull Model: Theory and Applications to Medical and Engineering Data
## 6 Applications
### Waiting Times of Service of 100 Bank Customers  
The data set given in Table 13 represents the waiting times (in minutes) before service of 100 bank customers (see [54]).  
Table 14, show that the distribution of this data set is positively skewed. By using equations Equations (22)-(25) to obtain the MLEs of the model parameters and the results are reported in Table 15.  
The estimated density with histogram of waiting times data for \(TLMW\) distribution and other distributions are displayed in Figure 17. The estimated cumulative and the fitted \(TLMW\) distribution and other distributions for the waiting times data set are displayed in Figure 18. The estimated PP and QQ plot for \(TLMW\) distribution are displayed for waiting times data in Figure 19.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline
0.8 & 0.8 & 1.3 & 1.5 & 1.8 & 1.9 & 1.9 & 2.1 & 2.6 & 2.7 & 2.9 & 3.1 & 3.2  
Figure 17: Pfd for different models for waiting times.  
Figure 18: Cdf for different models for waiting times.  
Figure 19: PP and QQ for \(TLMW\) model for waiting times.  
Now, applying the formal goodness-of-fit tests in order to verify which distribution fits better to these data set. The measures AKIC, BIC, CAKIC, HQIC and the KSD with PVKS (See Table 16) are considered.  
The 95% CIs for the parameters \(\alpha\), \(\beta\), \(\theta\) and \(\lambda\) are presented for MLE and Bayesian estimation methods in Table 17. Figure 20 displayed contour plot of parameters for \(TLMW\) model to check uniqueness and max-value of likelihood for waiting times. While, Figure 21 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 waiting times. Figure 22 discussed ACF test. Auto-correlation for MCMC is the correlation between a iteration series with a lagged version of itself. The ACF starts at a lag of 0, which is the correlation of the iteration series with itself and therefore results in a correlation of 1.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Models** & **AKIC** & **BIC** & **CAKIC** & **HQIC** & **KSD** & **PVKS** \\ \hline \(TLMW\) & 642.067 & 652.488 & 642.489 & 646.285 & 0.036 & 0.999 \\ KW & 641.950 & 652.371 & 642.371 & 646.167 & 0.038 & 0.999 \\ GMW & 642.387 & 652.808 & 642.808 & 646.605 & 0.041 & 0.996 \\ EOWL & 642.231 & 652.652 & 642.652 & 646.449 & 0.038 & 0.999 \\ WL & 642.533 & 652.954 & 642.954 & 646.751 & 0.041 & 0.995 \\ MOAPEW & 652.324 & 665.350 & 652.963 & 657.596 & 0.082 & 0.509 \\ EGAPEx & 642.184 & 652.604 & 642.605 & 646.401 & 0.040 & 0.997 \\ \hline \hline \end{tabular}
\end{table}
Table 16: Discrimination criteria and KS test of the \(TLMW\) model parameters for waiting times.  
Figure 20: Contour plot for parameters of \(TLMW\) model for waiting times.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & & \(\alpha\) & \(\beta\) & \(\theta\) & \(\lambda\) \\ \hline \multirow{4}{*}{MLE} & estimates & 0.0253 & 0.0905 & 0.8581 & 2.7331 \\  & SE & 0.6659 & 0.4810 & 1.9559 & 3.3520 \\  & Lower & 0.0005 & 0.0419 & 0.4747 & 0.7041 \\  & upper & 1.3305 & 1.0334 & 4.6917 & 9.3031 \\ \hline \multirow{4}{*}{Bayes} & estimates & 0.0252 & 0.0903 & 0.8580 & 2.7328 \\  & SE & 0.0050 & 0.0158 & 0.0652 & 0.1118 \\ \cline{1-1}  & Lower & 0.0155 & 0.0608 & 0.7276 & 2.5096 \\ \cline{1-1}  & upper & 0.0347 & 0.1227 & 0.9873 & 2.9423 \\ \hline \hline \end{tabular}
\end{table}
Table 17: MLE and Bayesian estimation methods of the \(TLMW\) model parameters for waiting times.  
Figure 21: MCMC plots for parameters of \(TLMW\) model for waiting times.