# An In-Depth Review of the Weibull Model with a Focus on Various Parameterizations
## 7 Conclusions and Future Work  
In this paper, a general parameterization for Weibull regression models was introduced, based on central tendency measures and shape parameters. A comprehensive review of Weibull regression models was provided, specifically focusing on different parameterizations of the Weibull distribution that utilize central tendency measures. The interpretation of regression coefficients when incorporating regression structures into these  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{2}{c}{**WEI2**} \\  & **Parameters** & **Estimate** & **SE** \\ \hline \multirow{3}{*}{\(\lambda\)} & Intercept & \(-20.5374\) & \(1.1339\) \\  & residual & \(0.9720\) & \(0.4152\) \\  & age & \(0.1401\) & \(0.0207\) \\ \cline{2-4}  & Intercept & \(0.5559\) & \(0.0155\) \\ \hline \multicolumn{2}{c}{AIC} & & \(186.04\) \\ \multicolumn{2}{c}{KS \(p\)-value} & & \(0.6703\) \\ \multicolumn{2}{c}{AD \(p\)-value} & & \(0.1916\) \\ \multicolumn{2}{c}{CVM \(p\)-value} & & \(0.1895\) \\ \hline \hline \end{tabular}
\end{table}
Table 10: ML estimate, standard error (SE) for WEI2, considering the residual plus age covariates for the ovarian dataset.  
Figure 5: QQ plot with the envelope of RQ residuals for the ovarian dataset. Note that the plot is identical for WEI, WEI2, and WEI3 models.  
Figure 6: Different measures of central tendency for the survival time in the ovarian problem in terms of age: residual = 1 (**left panel**) and residual = 2 (**right panel**). The plots are identical for WEI, WEI2, and WEI3 models.  
parameterizations was also explored. Furthermore, closed-form expressions for the expected Fisher information matrix have been derived for this general parameterization of Weibull regression models. Two types of residuals were explored, and maximum likelihood inference was implemented for estimating model parameters. The performance of this inference method was assessed through Monte Carlo simulations.  
A significant result, as established in Theorem 1, is that when modeling only \(\lambda\) in the WEI, WEI2, WEI3, WEI4, and WEI5 models, the same probability density function is yielded by all these parameterizations. Therefore, in these cases, it does not matter which model is used, as the conclusions for any measure of interest must be the same for any of the five models.  
To illustrate the practical relevance of the approach, two real-world applications using authentic datasets were presented. These applications underscored the adequacy of the introduced Weibull models when data present an asymmetric distribution. Depending on which parameter is of interest in the study, one model will be more convenient to use than the other.  
As part of future research, the plan is to develop an R package that facilitates inference in WEI4 and WEI5 regression models.  
Conceptualization, D.I.G., Y.M.G. and M.B.; methodology, D.I.G., Y.M.G. and M.B.; software, D.I.G., Y.M.G., C.M., L.S. and M.B.; validation, formal analysis, D.I.G., Y.M.G., C.M., L.S. and M.B.; investigation, D.I.G., Y.M.G., C.M., L.S. and M.B.; resources; data curation, Y.M.G. and C.M.; writing--original draft preparation, D.I.G., Y.M.G., C.M., L.S. and M.B.; writing--review and editing, D.I.G., Y.M.G., C.M., L.S. and M.B.; visualization, D.I.G., Y.M.G., C.M. and L.S. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding  
Data Availability Statement: Data and computational codes are available upon request from the authors.  
The authors declare no conflicts of interest.