# Topp-Leone Modified Weibull Model: Theory and Applications to Medical and Engineering Data
## 3 Some Statistical Features
### Moments and Moment Generating Function  
The \(r_{th}\) M of \(TLMW\) distribution can be computed using the following expression \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Parameters**} & \multicolumn{3}{c}{**Quantiles**} \\ \hline \(\alpha\) & \(\beta\) & \(\theta\) & \(\lambda\) & Median & Skewness (\(B_{s}\)) & Kurtosis (\(M_{k}\)) \\ \hline
0.6 & 0.5 & 1.0 & 0.70 & 0.753263 & \(-\)0.311058 & 6.59101 \\  & & & 1.0 & 0.667129 & \(-\)0.268271 & 5.19103 \\  & & & 2.0 & 0.445061 & \(-\)0.123186 & 3.49902 \\  & & & 7.0 & 0.0588123 & 0.49274 & 2.52725 \\
1.0 & 1.0 & 2.0 & 0.70 & 0.838 & \(-\)0.391413 & 6.32248 \\  & & & 1.0 & 0.77687 & \(-\)0.348819 & 4.99573 \\  & & & 2.0 & 0.603527 & \(-\)0.202834 & 3.3833 \\  & & & 7.0 & 0.17078 & 0.433166 & 2.46412 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Quantiles for \(TLMW\) distribution. \[{\mu_{r}}^{\prime} = \int_{0}^{\infty}x^{\prime}f(x,\varphi)dx=2\lambda\sum\limits_{i,j=0} ^{\infty}\pi(i,j)\int_{0}^{\infty}\Big{(}\alpha x^{r+\theta j}+\theta\beta x^{r+ \theta(j+1)-1}\Big{)}e^{-2\alpha(i+1)x}dx \tag{17}\] \[= 2\lambda\sum\limits_{i,j=0}^{\infty}\pi(i,j)\Bigg{[}\frac{ \alpha\Gamma(r+\theta j+1)}{[2\alpha(i+1)]^{r+\theta j+1}}+\theta\beta\frac{ \Gamma(r+\theta(j+1))}{[2\alpha(i+1)]^{r+\theta(j+1)}}\Bigg{]}.\]  
Further, the \(r_{th}\) central Ms \(\mu_{r}\) and cumulants \(\kappa_{r}\) of the \(TLMW\) distribution can be determined from \(\mu_{r}=\sum\limits_{m=0}^{r}(^{r}_{m})(-1)^{m}{\mu_{1}^{\prime}}^{m}\)\(\mu_{r-m}^{\prime}\) and \(\kappa_{r}=\mu_{r}^{\prime}-\sum\limits_{m=1}^{r-1}(^{r-1}_{m-1})\kappa_{m}\)\(\mu_{r-m}^{\prime}\), respectively, where \(\kappa_{1}=\mu_{1}^{\prime}\), \(\kappa_{2}=\mu_{2}^{\prime}-\mu_{1}^{\prime}\), \(\kappa_{3}=\mu_{3}^{\prime}-3\mu_{2}^{\prime}\)\(\mu_{1}^{\prime}+2\mu_{1}^{\prime 3}\), and \(\kappa_{4}=\mu_{4}^{\prime}-4\mu_{1}^{\prime}\mu_{3}^{\prime}-3\mu_{2}^{\prime 2}+12 \mu_{2}^{\prime 2}\mu_{1}^{\prime 2}-6\mu_{1}^{\prime 4}\), etc. Additionally, \(CS\) and \(CK\) can be obtained according to the following relation \(CS=\frac{\kappa_{3}}{\sqrt{\kappa_{3}^{2}}}\) and \(CK=\frac{\kappa_{4}}{\kappa_{2}^{2}}\), respectively. The MGF of \(TLMW\) model is provided as below  
\[M_{X}(t) = \int_{0}^{\infty}e^{tx}f(x,\varphi)dx=2\lambda\sum\limits_{i,j=0} ^{\infty}\pi(i,j)\Big{(}\alpha x^{\theta j}+\theta\beta x^{\theta(j+1)-1} \Big{)}e^{-(2\alpha(i+1)-t)x}dx \tag{18}\] \[= 2\lambda\sum\limits_{i,j=0}^{\infty}\pi(i,j)\Bigg{[}\frac{ \alpha\Gamma(\theta j+1)}{[2\alpha(i+1)-t]^{\theta j+1}}+\theta\beta\frac{ \Gamma(\theta(j+1))}{[2\alpha(i+1)-t]^{\theta(j+1)}}\Bigg{]}.\]  
Tables 2 and 3 show the numerical values of \({\mu_{1}}^{\prime}\), \({\mu_{2}}^{\prime}\), \({\mu_{3}}^{\prime}\), \({\mu_{4}}^{\prime}\), variance (V), skewness (CS) and kurtosis (CK) of the \(TLMW\) distribution. \begin{table}
\begin{tabular}{l l l l l l l l l} \hline \(\theta\) & \(\lambda\) & \({\mu_{1}}^{\prime}\) & \({\mu_{2}}^{\prime}\) & \({\mu_{3}}^{\prime}\) & \({\mu_{4}}^{\prime}\) & \(Variance\) & \(CS\) & \(CK\) \\ \hline  & 0.4 & 0.215 & 0.250 & 0.522 & 1.566 & 0.204 & 4.1