## 3 NGLog-Weib Distribution  
For a three-parameter special case of the NGLog-X family, let us consider the CDF\({}_{F}(x;\Theta)\), and PDF, \(f(x;\Theta)\), of the traditional Weib distribution are \(F(x;\Theta)=1-e^{-\beta x^{a}}\) and \(f(x;\Theta)=\beta\alpha x^{a-1}e^{-\beta x^{a}}\), respectively (\(\beta\in\mathbb{R}^{+},\alpha\in\mathbb{R}^{+}\), and \(x\in\mathbb{R}^{+}\)), where \(\Theta=(\beta,\alpha)\).  
Using \(F(x;\Theta)=1-e^{-\beta x^{a}}\) in Equation (7), we define the CDF \(G(x;\delta,\Theta)\) of the NGLog-Weib model. For \(x\in\mathbb{R}^{+}\), the scale parameter, \(\beta\in\mathbb{R}^{+}\), and shape parameters, \(\delta\in\mathbb{R}^{+}\) and \(\alpha\in\mathbb{R}^{+}\), the \(G(x;\delta,\Theta)\) CDF of the NGLog-Weib model is as follows,  
\[G(x;\delta,\Theta)=\frac{e^{\delta}\;\left(1-e^{-\beta x^{a}}\right)}{\left(e -\log\left(1-e^{-\beta x^{a}}\right)\right)^{\delta}}, \tag{13}\]  
\[S(x;\delta,\Theta)=1-\frac{e^{\delta}\;\Big{(}1-e^{-\beta x^{a}}\Big{)}}{\left( e-\log\left(1-e^{-\beta x^{a}}\right)\right)^{\delta}},\;\;\text{( Survival Function)} \tag{14}\]
Corresponding to \(G(x;\delta,\Theta)\) in Equation (13), the PDF\(g(x;\delta,\Theta)\), is  
\[g(x;\delta,\Theta)=\frac{\alpha\beta x^{\alpha-1}e^{-\beta x^{\alpha}+\delta} \bigg{\{}1+\delta\Big{(}e-\log\Big{(}1-e^{-\beta x^{\alpha}}\Big{)}\Big{)}^{-1} \bigg{\}}}{\big{(}e-\log\big{(}1-e^{-\beta x^{\alpha}}\big{)}\big{)}^{\delta}} \tag{15}\]  
For a graphical illustration, some possible behaviours of the PDF \(g(x;\delta,\Theta)\) of the NGLog-Weib distribution at different parameter values are sketched in Figure 2. The plots are sketched for (i) \(\delta=3.0\), \(\beta=3.5\), \(\alpha=0.3\) (in red), (ii) \(\delta=3.0\), \(\beta=5.0\), \(\alpha=1.9\) (in green), (iii) \(\delta=4.5\), \(\beta=3.0\), \(\alpha=2.9\) (in black), and (iv) \(\delta=6.5\), \(\beta=1.5\), \(\alpha=3.3\) (in blue). Figure 2 shows four different shapes of the PDF \(g(x;\delta,\Theta)\), including (i) decreasing or a reverse-J pattern (in red), (ii) right-skewed (in green), (iii) symmetrical (in black), and (iv) left-skewed (in blue).  
Furthermore, the hazard function \(h(x;\delta,\Theta)\), RHF (reverse hazard function) \(\tau(x;\delta,\Theta)\), and CHF (cumulative hazard function) \(H(x;\delta,\Theta)\) of the NGLog-Weib are given in Equations (16) to (18), respectively, by  
\[h(x;\delta,\Theta)=\frac{\alpha\beta x^{\alpha-1}e^{-\beta x^{\alpha}+\delta} \bigg{\{}1+\delta\Big{(}e-\log\Big{(}1-e^{-\beta x^{\alpha}}\Big{)}\Big{)}^{-1 }\bigg{\}}}{\big{(}e-\log\big{(}1-e^{-\beta x^{\alpha}}\big{)}\big{)}^{\delta }-e^{\delta}\big{(}1-e^{-\beta x^{\alpha}}\big{)}}, \tag{16}\]  
\[\tau(x;\delta,\Theta)=\frac{\beta\alpha x^{\alpha-1}e^{-\beta x^{\alpha}} \bigg{\{}1+\delta\Big{(}e-\log\Big{(}1-e^{-\beta x^{\alpha}}\Big{)}\Big{)}^{-1 }\bigg{\}}}{\big{(}1-e^{-\beta x^{\alpha}}\big{)}}, \tag{17}\]  
\[H(x;\delta,\Theta)=-\log\left(1-\frac{e^{\delta}\left(1-e^{-\beta x^{\alpha}} \right)}{\big{(}e-\log\big{(}1-e^{-\beta x^{\alpha}}\big{)}\big{)}^{\delta}}\right) \tag{18}\]Furthermore, for the various values of the parameters, some possible behaviours of the HF \(h(x;\delta,\Theta)\) of the NGLog-Weib model are shown in Figure 3 for (i) \(\delta=10.7\), \(\beta=2.6\), \(\alpha=0.5\) (in red), (ii) \(\delta=10.4\), \(\beta=3.5\), \(\alpha=0.5\) (in green), (iii) \(\delta=0.5\), \(\beta=0.2\), \(\alpha=2.9\) (in black), (iv) \(\delta=6.0\), \(\beta=2.4\), \(\alpha=0.3\) (in blue), and (iv) \(\delta=0.5\), \(\beta=0.8\), \(\alpha=0.5\) (in violet). From Figure 3, we can observe (or examine) that the NGLog-Weib offers data modelling with increasing, decreasing, uni-model, and bathtub-shaped HFs, \(h(x;\delta,\Theta)\).