*4.4. Stochastic Ordering*

In several areas of probability and statistics, stochastic ordering and disparities are being adhered to at an accelerating rate. For example, in analyzing the contrast of investment returns to random cash flows; two manufacturers may use distinct technologies to make gadgets with the same function, resulting in non-identical life distributions or comparing the strength of dependent structures. Here, we use the term stochastic ordering to refer to any ordering relation on a space of probability measures in a wide sense. Let *X* and *Y* be two rvs from OGE2Fr distributions, with assumptions previously mentioned in Section 3. Given that *a*1 < *a*2, and for *X*1 ≤*lr X*2, *f*1(*x*)/ *f*2(*x*) shall be decreasing in *x* if and only if the following result holds Let *X* and *Y* be two rvs from OGE2Fr distributions, with assumptions previously mentioned in Section 3. Given that *a*1 < *a*2, and for *X*1 ≤*lr X*2, *f*1(*x*)/ *f*2(*x*) shall be decreasing in *x* if the following result holds

$$\begin{split} \frac{d}{dx}\log\left[\frac{f\_1(\mathbf{x})}{f\_2(\mathbf{x})}\right] &= \quad \left(a\_2 - a\_1\right)b\,a^b\mathbf{x}^{-(b+1)}\mathbf{e}^{-(a/x)^b}\mathbf{1} - \mathbf{e}^{-(a/x)^b}\mathbf{1} - \mathbf{e}^{-(a/x)^b} \left(-\mathbf{e}^{-(a/x)^b}\right) \\ &\quad + (\beta - 1)\left[b\,a^b\mathbf{x}^{-(b+1)}\mathbf{e}^{-(a/x)^b}\right]\left(1 - \mathbf{e}^{-(a/x)^b}\right)^{-1} \\ &\quad \times \left[\frac{a\_1\left(1 - \mathbf{e}^{-(a/x)^b}\right)^{-a\_1}}{\mathbf{e}^{-\left(1 - \mathbf{e}^{-(a/x)^b}\right)^{b-1}} - 1}\right] \\ &\quad + (1 - \beta)\left[b\,a^b\mathbf{x}^{-(b+1)}\mathbf{e}^{-(a/x)^b}\right]\left\{1 - \mathbf{e}^{-(a/x)^b}\right\}^{-1} \\ &\quad \times \left[\frac{a\_1\left(1 - \mathbf{e}^{-(a/x)^b}\right)^{-a\_2}}{\mathbf{e}^{-\left(1 - \mathbf{e}^{-(a/x)^b}\right)^{-a\_2}} - 1}\right] < 0. \end{split}$$
