*4.2. Parameter Estimation*

Let *x*1, ... , *xn* be a sample of size *n* from the OGE2Fr distribution given in Equation (18). The log-likelihood function = (Θ) for the vector of parameters Θ = (*<sup>α</sup>*, *β*, *a*, *b*) is

$$\begin{split} \ell^{\ell} &= \ n \boldsymbol{b} \log(\boldsymbol{a}) + \boldsymbol{n} \log(\boldsymbol{a} \boldsymbol{\beta} \boldsymbol{b}) - (\boldsymbol{b} + 1) \sum\_{i=1}^{n} \log(\mathbf{x}\_{i}) - \sum\_{i=1}^{n} \left( \frac{\boldsymbol{a}}{\mathbf{x}\_{i}} \right)^{\boldsymbol{b}} + \sum\_{i=1}^{n} \left[ 1 - \left\{ 1 - \mathbf{e}^{-\left(\boldsymbol{a}/\boldsymbol{x}\_{i}\right)^{\boldsymbol{b}}} \right\}^{-\boldsymbol{a}} \right] \\ &- (\boldsymbol{a} + 1) \sum\_{i=1}^{n} \log \left[ 1 - \mathbf{e}^{-\left(\frac{\boldsymbol{a}}{\mathbf{x}\_{i}}\right)^{\boldsymbol{b}}} \right] + (\boldsymbol{\beta} - 1) \sum\_{i=1}^{n} \log \left[ 1 - \mathbf{e}^{-\left(1 - \frac{\boldsymbol{a} - \left(\frac{\boldsymbol{a}}{\mathbf{x}\_{i}}\right)^{\boldsymbol{b}}}\right)^{-\boldsymbol{a}}} \right]. \end{split} \tag{25}$$

The components of score vectors U(Θ) are

$$\begin{split} \mathcal{U}\_{n} &= \quad \frac{n}{\pi} - \sum\_{i=1}^{n} \log \left[ 1 - \mathbf{e}^{-\left(\frac{\mathbf{d}}{\hat{\pi}\_{i}}\right)^{b}} \right] + \sum\_{i=1}^{n} \left[ 1 - \mathbf{e}^{-\left(\frac{\mathbf{d}}{\hat{\pi}\_{i}}\right)^{b}} \right]^{-a} \log \left[ 1 - \mathbf{e}^{-\left(\frac{\mathbf{d}}{\hat{\pi}\_{i}}\right)^{b}} \right] \\ &+ (\beta - 1) \sum\_{i=1}^{n} \frac{\left\{ 1 - \mathbf{e}^{-\left(\frac{\mathbf{d}}{\hat{\pi}\_{i}}\right)^{b}} \right\}^{-a} \log \left\{ 1 - \mathbf{e}^{-\left(\frac{\mathbf{d}}{\hat{\pi}\_{i}}\right)^{b}} \right\}}{1 - \mathbf{e}^{-1 + \left\{ 1 - \mathbf{e}^{-\left(\frac{\mathbf{d}}{\hat{\pi}\_{i}}\right)^{b}} \right\}^{-a}}, \\ \mathcal{U}\_{\beta} &= \quad \frac{n}{\beta} + \sum\_{i=1}^{n} \log \left[ 1 - \mathbf{e}^{-\left(\frac{\mathbf{d}}{\hat{\pi}\_{i}}\right)^{b}} \right], \end{split}$$

$$\begin{split} \mathcal{U}\_{a} &= \quad \frac{nb}{a} - \frac{a^{b}b}{a} \sum\_{i=1}^{n} \mathbf{x}\_{i}^{-b} - \frac{a^{b}b(a+1)}{a} \sum\_{i=1}^{n} \frac{\mathbf{x}\_{i}^{-b}}{\mathbf{e} \left(\frac{\mathbf{x}}{\mathbf{x}\_{i}}\right)^{b} - 1} - \frac{a^{b}ba}{a} \sum\_{i=1}^{n} \frac{\mathbf{x}\_{i}^{-b} \left\{1 - \mathbf{e}^{-\left(\frac{a}{\mathbf{x}\_{i}}\right)^{b}}\right\}^{-a}}{1 - \mathbf{e}^{-\left(\frac{a}{\mathbf{x}\_{i}}\right)^{b}}} \\ &+ \frac{a(\beta - 1) \, a^{b} \, b}{a} \sum\_{i=1}^{n} \frac{\mathbf{x}\_{i}^{-b} \left\{1 - \mathbf{e}^{-\left(\frac{a}{\mathbf{x}\_{i}}\right)^{b}}\right\}^{-a}}{\left[1 - \mathbf{e}^{\left(\frac{a}{\mathbf{x}\_{i}}\right)^{b}}\right] \left[1 - \mathbf{e}^{-1 + \left\{1 - \mathbf{e}^{-\left(\frac{a}{\mathbf{x}\_{i}}\right)^{b}}\right\}^{-a}}\right]}, \end{split}$$

$$\begin{split} \mathcal{U}\_{b} &= \quad \frac{n}{b} + n \log(a) - \sum\_{i=1}^{n} \log(\mathbf{x}\_{i}) - a^{b} \sum\_{i=1}^{n} \mathbf{x}\_{i}^{-b} \log\left(\frac{a}{\mathbf{x}\_{i}}\right) \\ &- (a+1) \, a^{b} \sum\_{i=1}^{n} \frac{\mathbf{x}\_{i}^{-b} \log\left(\frac{a}{\mathbf{x}\_{i}}\right)}{\mathbf{e}\left(\frac{a}{\mathbf{x}\_{i}}\right)^{b} - 1} + a \, a^{b} \sum\_{i=1}^{n} \frac{\mathbf{x}\_{i}^{-b} \log\left(\frac{a}{\mathbf{x}\_{i}}\right) \left\{1 - \mathbf{e}^{-\left(\frac{a}{\mathbf{x}\_{i}}\right)^{b}}\right\}^{-a-b}}{\mathbf{e}\left(\frac{a}{\mathbf{x}\_{i}}\right)^{b} - 1} \\ &+ a \, a^{b} (\beta - 1) \sum\_{i=1}^{n} \frac{\mathbf{x}\_{i}^{-b} \log\left(\frac{a}{\mathbf{x}\_{i}}\right) \left\{1 - \mathbf{e}^{-\left(\frac{a}{\mathbf{x}\_{i}}\right)^{b}}\right\}^{-a}}{\left[\mathbf{e}\left(\frac{a}{\mathbf{x}\_{i}}\right)^{b} - 1\right] \left[1 - \mathbf{e}^{-1 + \left(1 - \mathbf{e}^{-\left(\frac{a}{\mathbf{x}\_{i}}\right)^{b}}\right)^{-a}}\right]}. \end{split}$$
