# An In-Depth Review of the Weibull Model with a Focus on Various Parameterizations
## 5 Monte Carlo Simulation Studies
### MC Simulation Study II: Residuals Analysis  
We delve into the analysis of the Cox-Snell (CS) and randomized quantile (RQ) residuals for the WEI4 and WEI5 models. The computation of the CS residual is as follows:  
\[\tilde{r}_{i}^{\text{CS}}=-\log(\widehat{S}(y_{i};\mathbf{x}_{i})),\quad i=1,\ldots,n,\]  
where \(\widehat{S}(y_{i}|\mathbf{x}_{i})\) is the estimated survival function obtained from the fitted model with values \(\mathbf{x}_{i}\) for the explanatory variables. The survival functions of the models under consideration are presented in Section 2. The RQ residual is defined as  
\[\tilde{r}_{i}^{\text{RQ}}=\Phi^{-1}(\widehat{S}(y_{i}|\mathbf{x}_{i})),\quad i=1, \ldots,n,\]  
where \(\Phi^{-1}\) is the quantile function of the standard normal distribution. For this investigation, we maintain the same simulation scenario as in the MC simulation study I, as outlined above. Tables 5 and 6 display the empirical mean, standard deviation (SD), and coefficient of skewness (CSk) of the CS and RQ residuals. Theoretical expectations for these statistics are 1, 1, and 2, respectively, for the CS residuals, and 0, 1, and 0, respectively, for the RQ residuals, due to their respective theoretical distributions being exponential and standard normal. The results in Tables 5 and 6 demonstrate that, in general, both types of residuals provide good approximations for the WEI4 and WEI5 models. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{\(\beta_{1}=0.5;\quad\beta_{2}=1.5;\quad\zeta_{1}=0.2;\quad\zeta_{2}=1.0\)} \\ \cline{2-9}  & \multicolumn{4}{c}{Bias} & \multicolumn{4}{c}{MSE} \\ \hline \(n\) & \(\beta_{1}\) & \(\beta_{2}\) & \(\zeta_{1}\) & \(\zeta_{2}\) & \(\beta_{1}\) & \(\beta_{2}\) & \(\zeta_{1}\) & \(\zeta_{2}\) \\ \hline
30 & \(-0.0136\) & \(0.0217\) & \(0.0205\) & \(0.1142\) & \(0.4725\) & \(1.5218\) & \(0.0842\) & \(0.3431\) \\
50 & \(0.0013\) & \(0.0040\) & \(0.0104\) & \(0.0720\) & \(0.0886\) & \(0.2640\) & \(0.0414\) & \(0.1756\) \\
100 & \(-0.0034\) & \(0.0078\) & \(0.0056\) & \(0.0324\) & \(0.0485\) & \(0.1301\) & \(0.0226\) & \(0.0738\) \\
500 & \(0.0004\) & \(-0.0003\) & \(0.0010\) & \(0.0063\) & \(0.0062\) & \(0.0169\) & \(0.0027\) & \(0.0108\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._ \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{4}{c}{\(\beta_{1}=0.2;\quad\beta_{2}=1.0;\quad\zeta_{1}=0.2;\quad\zeta_{2}=1.0\)} \\ \cline{2-7}  & \multicolumn{4}{c}{Randomized Quantile} & \multicolumn{4}{c}{Cox–Snell} \\ \hline \(n\) & Mean & SD & CSk & Mean & SD & CSk \\ \hline
30 & \(-0.0176\) & \(1.0197\) & \(0.0225\) & \(0.9879\) & \(0.9686\) & \(1.2599\) \\
50 & \(-0.0064\) & \(1.0067\) & \(0.0761\) & \(0.9936\) & \(1.0119\) & \(1.8385\) \\
100 & \(0.0057\) & \(0.9918\) & \(0.1559\) & \(1.0024\) & \(1.0380\) & \(1.9373\) \\
500 & \(0.0085\) & \(0.9903\) & \(0.1282\) & \(1.0057\) & \(1.0256\) & \(1.9420\) \\ \hline \hline  & \multicolumn{4}{c}{\(\beta_{1}=0.2;\quad\beta_{2}=0.5;\quad\zeta_{1}=0.2;\quad\zeta_{2}=1.0\)} \\ \hline \multicolumn{7}{c}{randomized quantile} & \multicolumn{4}{c}{Cox–Snell} \\ \hline \(n\) & mean & SD & CSk & mean & SD & CSk \\ \hline
30 & \(-0.0124\) & \(0.9931\) & \(0.3132\) & \(0.9862\) & \(1.0388\) & \(1.4458\) \\
50 & \(-0.0045\) & \(1.0443\) & \(-0.3680\) & \(1.0011\) & \(0.9207\) & \(1.3308\) \\
100 & \(-0.0268\) & \(1.0082\) & \(-0.0545\) & \(0.9732\) & \(1.0482\) & \(2.7658\) \\
500 & \(-0.0038\) & \(0.9844\) & \(0.2264\) & \(0.9942\) & \(1.0366\) & \(2.0337\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Mean, SD, and CSk of the CS and randomized quantile residuals for the WEI4 model, with the indicated sample size and true values of the parameters.