*3.7. Stochastic Ordering*

Stochastic ordering has indeed been acknowledged as an essential tool for assessing comparative behavior in reliability theory and other disciplines. Assume *X* and *Y* be two rvs via cdfs, sfs and pdfs *<sup>F</sup>*1(*x*) and *<sup>F</sup>*2(*x*), *<sup>F</sup>*¯1(*x*) = 1 − *<sup>F</sup>*1(*x*) and *<sup>F</sup>*¯2(*x*) = 1 − *<sup>F</sup>*2(*x*), and *f*1(*x*) and *f*2(*x*), respectively. In the specific planning, the rv *X*1 is considered to be lower than *X*2:


All these four stochastic orders studied in (1)–(4) are connected to one another as a result of [37] and the accompanying ramifications apply:

$$(X\_1 \le\_{\text{rhr}} X\_2) \Leftarrow (X\_1 \le\_{\text{lr}} X\_2) \Rightarrow (X\_1 \le\_{\text{lr}} X\_2) \Rightarrow (X\_1 \le\_{\text{st}} X\_2) \dots$$

when sufficient conditions are met, the OGE2-G distributions are ordered with regard to the strongest LL ratio ordering, as shown by the next theorem.

**Theorem 3.** *Assume X*1 ∼ *OGE2*(*<sup>α</sup>*1, *β*; *ψ*) *and X*2 ∼ *OGE2*(*<sup>α</sup>*2, *β*; *ψ*)*. If α*1 < *α*2*, then X*1 ≤*lr X*2.

**Proof.** First, we have the ratio

$$\frac{f\_1(\mathbf{x})}{f\_2(\mathbf{x})} = \frac{\alpha\_1 \beta \, k(\mathbf{x}) \overline{\mathbf{K}}^{-\mathbf{a}\_1 - 1} \mathbf{e}^{-\left\{\frac{1 - \overline{\mathbf{K}}^{\mathbf{a}\_1}}{\overline{\mathbf{K}}^{\mathbf{a}\_1}}\right\} \left[1 - \mathbf{e}^{-\left\{\frac{1 - \overline{\mathbf{K}}^{\mathbf{a}\_1}}{\overline{\mathbf{K}}^{\mathbf{a}\_1}}\right\} \right]^{\beta - 1}}{\alpha\_2 \beta \, k(\mathbf{x}) \overline{\mathbf{K}}^{-\mathbf{a}\_2 - 1} \mathbf{e}^{-\left\{\frac{1 - \overline{\mathbf{K}}^{\mathbf{a}\_2}}{\overline{\mathbf{K}}^{\mathbf{a}\_2}}\right\} \left[1 - \mathbf{e}^{-\left\{\frac{1 - \overline{\mathbf{K}}^{\mathbf{a}\_2}}{\overline{\mathbf{K}}^{\mathbf{a}\_2}}\right\} \right]^{\beta - 1}}.$$

After simplification, we obtain

$$\frac{f\_1(\mathbf{x})}{f\_2(\mathbf{x})} = \frac{\alpha\_1 \mathbb{E}^{(\alpha\_2 - \alpha\_1)} \mathbf{e}^{-\left\{\frac{1 - \mathbb{E}^{\alpha\_1}}{\mathbb{E}^{\alpha\_1}}\right\} + \left\{\frac{1 - \mathbb{E}^{\alpha\_2}}{\mathbb{E}^{\alpha\_2}}\right\} \left[1 - \mathbf{e}^{-\left\{\frac{1 - \mathbb{E}^{\alpha\_1}}{\mathbb{E}^{\alpha\_1}}\right\}}\right]^{\beta - 1}}{\alpha\_2 \left[1 - \mathbf{e}^{-\left\{\frac{1 - \mathbb{E}^{\alpha\_1}}{\mathbb{E}^{\alpha\_1}}\right\}}\right]^{\beta - 1}}.$$

Next,

$$\begin{split} \log\left[\frac{f\_1(\mathbf{x})}{f\_2(\mathbf{x})}\right] &=& -\log(a\_2 - a\_1) + (a\_2 - a\_1)\log(\overline{\mathcal{K}}) - \left\{\frac{1 - \overline{\mathcal{K}}^{a\_1}}{\overline{\mathcal{K}}^{a\_1}}\right\} + \left\{\frac{1 - \overline{\mathcal{K}}^{a\_2}}{\overline{\mathcal{K}}^{a\_2}}\right\} \\ &+ (\beta - 1)\log\left[1 - \mathbf{e}^{-\left\{\frac{1 - \overline{\mathcal{K}}^{a\_1}}{\overline{\mathcal{K}}^{a\_1}}\right\}}\right] \\ &- (\beta - 1)\log\left[1 - \mathbf{e}^{-\left\{\frac{1 - \overline{\mathcal{K}}^{a\_2}}{\overline{\mathcal{K}}^{a\_2}}\right\}}\right]. \end{split}$$

If *a*1 < *a*2, we obtain

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

Thus, *f*1(*x*)/ *f*2(*x*) is decreasing in *x* and hence *X*1 ≤*lr X*2.
