**4. OGE2-Fréchet Distribution**

In this section, we study the first special model defined in Section 2.2, the OGE2- Fréchet (OGE2Fr), in view of its practical application.

The OGE2Fr model can be defined from (1) by taking *<sup>K</sup>*(*x*; *ψ*) = e<sup>−</sup>(*a*/*x*)*<sup>b</sup>* and *k*(*x*; *ψ*) = *b abx*−(*b*+<sup>1</sup>)e<sup>−</sup>(*a*/*x*)*<sup>b</sup>* , as cdf and pdf of the baseline Fréchet distribution with *a*, *b* > 0, respectively. The cdf and pdf of OGE2Fr distribution are, respectively, given by

$$F(\mathbf{x}; \boldsymbol{a}, \boldsymbol{\beta}, a, b) = \left[ 1 - \mathbf{e}^{1 - \left\{ 1 - \mathbf{e}^{-\left(\frac{\mathbf{x}}{\boldsymbol{\beta}}\right)^{b}} \right\}^{-a}} \right]^{\boldsymbol{\beta}}, \mathbf{x} > 0 \quad \boldsymbol{a}, \boldsymbol{\beta}, a, b > 0,\tag{17}$$

and

$$f(\mathbf{x}; a, \boldsymbol{\upbeta}, a, b) \quad = \quad a^b \tag{18} \\ \boldsymbol{a} \cdot \boldsymbol{b} \,\boldsymbol{\upbeta} \mathbf{x}^{-b-1} \mathbf{e}^{1-\left(\frac{\mathbf{x}}{x}\right)^b - \left(1 - \mathbf{e}^{-\left(\frac{\mathbf{x}}{x}\right)^b}\right)^{-a}} \left(1 - \mathbf{e}^{-\left(\frac{\mathbf{x}}{x}\right)^b}\right)^{-a-1}$$

$$\times \left[1 - \mathbf{e}^{1-\left(1 - \mathbf{e}^{-\left(\frac{\mathbf{x}}{x}\right)^b}\right)^{-a}}\right]^{\beta - 1} \,\tag{18}$$

where *α*, *β* and *b* are shape parameters while *a* is scale parameter. ThehrfandqfoftheOGE2Frdistributionareobtainedas

$$h(\mathbf{x}) = \left. a^b a b \,\beta \mathbf{x}^{-b-1} \mathbf{e}^{1-\left(\frac{\mathbf{e}}{\mathbf{z}}\right)^b - \left\{1 - \mathbf{e}^{-\left(\frac{\mathbf{e}}{\mathbf{z}}\right)^b}\right\}^{-a}} \left[1 - \mathbf{e}^{-\left(\frac{\mathbf{e}}{\mathbf{z}}\right)^b}\right]^{-a-1}$$

$$\times \left[1 - \mathbf{e}^{1-\left\{1 - \mathbf{e}^{-\left(\frac{\mathbf{e}}{\mathbf{z}}\right)^b}\right\}^{-a}}\right]^{1-\beta} \left[1 - \left\{1 - \mathbf{e}^{-\left(1 - \mathbf{e}^{-\left(\frac{\mathbf{e}}{\mathbf{z}}\right)^b}\right)^{-a}}\right\}^{\beta}\right] \tag{19}$$

$$Q(u) = \left[ a \left\{ -\log \left( 1 - \left[ 1 - \log \left( 1 - u^{1/\beta} \right) \right]^{-1/a} \right) \right\} \right]^{-1/b}.\tag{20}$$

Figure 1 depicts a visualisation of the pdf and hrf functions, exhibiting the range of shapes that all these functions can take at random input parametric values. The OGE2Fr distribution's pdf can be gradually decreasing, unimodal, and right-skewed, with different curves, tail, and asymmetric aspects, as shown in Figure 1. The hrf, on the other hand, offers an extensive range of increasing, decreasing, unimodal, and increasing-decreasingincreasing (IDI) forms. Given a wide variety of hrf shapes being offered, the OGE2Fr distribution can in fact be a useful tool to model unpredictable time-to-event phenomena.

### *4.1. Linear Representation and Related Properties*

The cdf of the OGE2Fr distribution is quite straightforward and is achieved by using the result defined in Equation (6) as

$$F(\mathbf{x}) = F(\mathbf{x} : \mathbf{a}, \boldsymbol{\beta}, \mathbf{a}, \boldsymbol{b}) = \sum\_{\ell=0}^{\infty} \xi\_{\ell}^{\mathbf{x}} \left[ \mathbf{e}^{-(\mathbf{a}/\mathbf{x})^{\mathbf{b}}} \right]^{\ell} \qquad \ell \ge 1,\tag{21}$$

where is the power parameter and noting that ∑∞=<sup>0</sup> *ξ* is unity. Forsimplicity,wecanrewritetheaboveresultas

$$F(\mathbf{x}) = F(\mathbf{x} : \mathbf{a}, \boldsymbol{\beta}, \mathbf{a}, \mathbf{b}) = \sum\_{\ell=0}^{\infty} \xi\_{\ell}^{\mathbf{x}} \mathbf{e}^{-\ell (\mathbf{a}/\mathbf{x})^{\mathbf{b}}}$$

By differentiating the last term, we can express the density of OGE2Fr model as follows

$$f(\mathbf{x} : \boldsymbol{a}, \boldsymbol{\beta}, \mathbf{a}, \boldsymbol{b}) = \sum\_{i,j=1}^{\infty} \xi\_{\ell}^{\mathrm{r}} \Pi(\mathbf{x}; \ell, \mathbf{a}, \mathbf{b}) \,, \tag{22}$$

.

where <sup>Π</sup>(*x*; , *a*, *b*) = *b abx*−*b*−1e−( *ax* )*b* represents the Fréchet density function with power parameter . Equation (22) enforces the fact that OGE2Fr density is a linear combination of Fréchet densities. Thus, we can derive various mathematical properties using Fréchet distribution.

**Figure 1.** (**a**) Plots of density and (**b**) hazard rate of the OGE2Fr model for some random parameter values.

Moments are the heart and soul of any statistical analysis. Moments can be used to evaluate the most essential characteristics such as mean, variance, skewness, and kurtosis of a distribution. We now directly present the mathematical expressions for the moments of OGE2Fr model as follows. Let *Y* be a random variable with density <sup>Π</sup>(*x*; , *a*, *b*). Then, core properties of *X* can follow from those of *Y*. First, the *p*th ordinary moment of *X* can be written as

$$
\mu\_p' = \sum\_{i,j=1}^{\infty} \xi\_\ell \ell \, a^p \, \frac{\ell^{1-p/b}}{\Gamma(1-p/b)}.\tag{23}
$$

Second, the cumulants (*<sup>κ</sup>p*) of *X* can be determined recursively from (23) as *κs* = *μs* − ∑*<sup>s</sup>*−<sup>1</sup> *k*=1 (*<sup>s</sup>*−<sup>1</sup> *<sup>k</sup>*−<sup>1</sup>) *κk <sup>μ</sup>s*−*k*, respectively, where *κ*1 = *<sup>μ</sup>*1. The skewness *γ*1 = *κ*3/*κ*3/2 2 and kurtosis *γ*2 = *κ*4/*κ*22 of *X* can be calculated from the third and fourth standardized cumulants. Plots of mean, variance, skewness and kurtosis of the OGE2Fr distribution are displayed in Figure 2. These plots signifies the significant role of the parameters *α* and *β* in modeling the behaviors of *X*.

Third, the *p*th incomplete moment of *X*, denoted by *mp*(*y*) = *E*(*Xp* | *X* ≤ *y*) = *y*0 *xp fOGE*2*Fr*(*x*)*dx*, is easily found changing variables from the lower incomplete gamma function *<sup>γ</sup>*(*<sup>v</sup>*, *u x*) = ∞0 *xv*−<sup>1</sup> e<sup>−</sup><sup>u</sup> xdx when calculating the corresponding moment of *Y*. Then, we obtain

$$m\_p(y) = \sum\_{i,j=1}^{\infty} \vec{\xi}\_{\ell} a^p \, ^t \ell^{1+p/b} \Gamma \Big( 1 - p/b, \ell \, (a/x)^b \Big). \tag{24}$$

Fourth, the first incomplete moment *<sup>m</sup>*1(*z*) is used to construct the Bonferroni and Lorenz curves as discussed in Section 3.2. Figure 3 provides the income inequality curves (Bonferroni & Lorenz) of the proposed distribution which can easily be derived from (24), respectively, where *q* = *Q*(*<sup>π</sup>*) is the qf of *X* derived from Equation (20).

**Figure 2.** Plots of (**a**) mean, (**b**) variance, (**c**) skewness and (**d**) kurtosis of the OGE2Fr distribution.

**Figure 3.** Plots of (**a**) Bonferroni curve and (**b**) Lorenz curve of OGE2Fr model.
