*4.6. Order Statistics*

Let *X*1, *X*2, ... , *Xn* be a random sample from the GOLE-F family with CDF and PDF defined in Equations (3) and (4), respectively. Suppose *X*1:*n*, *X*2:*n*, ... , *Xn*:*<sup>n</sup>* denote the order statistics obtained from this sample and *Xr*:*<sup>n</sup>* is the *i*th order statistic, then the density function of the *r*th order statistic is provided by

$$\log\_{r:n}(\mathbf{x}) = \frac{n!}{(r-1)!(n-r)!} \sum\_{s=0}^{n-r} (-1)^s \binom{n-r}{s} \mathbf{g}(\mathbf{x}) G(\mathbf{x})^{r+s-1}.\tag{26}$$

From (17), we determine

$$\left[G(\mathbf{x})^{r+s-1} = \left[\sum\_{m=0}^{\infty} \delta\_m F(\mathbf{x})^m\right]^{r+s-1} = \sum\_{m=0}^{\infty} d\_{r+s-1,m} F(\mathbf{x})^m,\tag{27}$$

where

$$d\_{r+s-1,0} = \delta\_0^{r+s-1} \text{ and } d\_{r+s-1,m} = (m\delta\_0)^{-1} \sum\_{q=1}^{m} [q(r+s) - m] \delta\_q d\_{r+s-1,m-q}.$$

By replacing *t* instead of *m* in Equation (19), we obtain

$$\mathbf{g}(\mathbf{x}) = \sum\_{t=0}^{\infty} \delta\_{t+1}(t+1) f(\mathbf{x}) F(\mathbf{x})^t. \tag{28}$$

By substituting (26) in (27) and (28), we determine the PDF of the *r*th order statistic *Xr*:*n* as

$$g\_{r:n}(\mathbf{x}) = \sum\_{t,m=0}^{\infty} \pi\_{t,m} h\_{t+m+1}(\mathbf{x}),\tag{29}$$

where *ht*+*m*+1(*x*) denotes the PDF of exp-F distribution with power parameter (*t* + *m* + 1), and

$$\pi\_{t,m} = \sum\_{s=0}^{n-r} \frac{(-1)^s \binom{n-r}{s} n! \delta\_{t+1}(t+1) d\_{r+s-1,m}}{(r-1)!(n-r)!(t+m+1)} \dots$$

Based on Equation (29), several mathematical properties of these order statistics such as ordinary and incomplete moments, factorial moments, moment generating function, mean deviations and several others, can be obtained.

#### *4.7. Stochastic Orderings*

Stochastic orders and inequalities are used in many different areas of probability and statistics. Such areas include reliability theory, survival analysis, economics, insurance, actuarial science, queuing theory, biology, operations research, management science, etc. For more detail regarding stochastic ordering, see (Shaked et al., 1994 [15]). Given two random variables *X* and *Y*, we say that *X* is smaller than *Y* in the:


For all the previous orders, we determine the following chains of implications:

$$X \leq\_{lr} Y \Rightarrow X \leq\_{lr} Y \Rightarrow X \leq\_{st} Y$$

and

$$X \leq\_{lr} Y \Rightarrow X \leq\_{rh} Y \Rightarrow X \leq\_{st} Y,$$

also

$$X \leq\_{lr} \mathcal{Y} \Rightarrow X \leq\_{surl} \mathcal{Y}.$$

For the proposed GOLE-F family, the following theorem provides the stochastic comparison results with respect to the above orderings.

**Theorem 1.** *Let X* ∼ *GOLE*(*a*1, *b*1, *c*1, *φ*) *and Y* ∼ *GOLE*(*a*2, *b*2, *c*2, *φ*)*. If a*<sup>1</sup> ≥ *a*2*. and b*<sup>1</sup> ≥ *b*<sup>2</sup> *and c*<sup>1</sup> ≤ *c*2*, then X* ≤*st Y.*

**Proof.** If *c*<sup>1</sup> ≤ *c*2, then

$$\frac{F(\mathbf{x})^{\varepsilon\_1}}{1 - F(\mathbf{x})^{\varepsilon\_1}} \ge \frac{F(\mathbf{x})^{\varepsilon\_2}}{1 - F(\mathbf{x})^{\varepsilon\_2}}.\tag{30}$$

Hence, if *a*<sup>1</sup> ≥ *a*<sup>2</sup> and *b*<sup>1</sup> ≥ *b*2, then

$$\frac{a\_1 F(\mathbf{x})^{\varepsilon\_1}}{1 - F(\mathbf{x})^{\varepsilon\_1}} + \frac{b\_1}{2} \left( \frac{F(\mathbf{x})^{\varepsilon\_1}}{1 - F(\mathbf{x})^{\varepsilon\_1}} \right)^2 \ge \frac{a\_2 F(\mathbf{x})^{\varepsilon\_2}}{1 - F(\mathbf{x})^{\varepsilon\_2}} + \frac{b\_2}{2} \left( \frac{F(\mathbf{x})^{\varepsilon\_2}}{1 - F(\mathbf{x})^{\varepsilon\_2}} \right)^2. \tag{31}$$

Therefore,

$$\left[ -\left( \frac{a\_1 F(\mathbf{x})^{c\_1}}{1 - F(\mathbf{x})^{c\_1}} + \frac{b\_1}{2} \left( \frac{F(\mathbf{x})^{c\_1}}{1 - F(\mathbf{x})^{c\_1}} \right)^2 \right) \right] \le \left[ -\left( \frac{a\_2 F(\mathbf{x})^{c\_2}}{1 - F(\mathbf{x})^{c\_2}} + \frac{b\_2}{2} \left( \frac{F(\mathbf{x})^{c\_2}}{1 - F(\mathbf{x})^{c\_2}} \right)^2 \right) \right].\tag{32}$$

Thus,

$$\begin{split} & 1 - \exp\left[ -\left( \frac{a\_1 F(\mathbf{x})^{\varepsilon\_1}}{1 - F(\mathbf{x})^{\varepsilon\_1}} + \frac{b\_1}{2} \left( \frac{F(\mathbf{x})^{\varepsilon\_1}}{1 - F(\mathbf{x})^{\varepsilon\_1}} \right)^2 \right) \right] \\ & \geq 1 - \exp\left[ -\left( \frac{a\_2 F(\mathbf{x})^{\varepsilon\_2}}{1 - F(\mathbf{x})^{\varepsilon\_2}} + \frac{b\_2}{2} \left( \frac{F(\mathbf{x})^{\varepsilon\_2}}{1 - F(\mathbf{x})^{\varepsilon\_2}} \right)^2 \right) \right]. \end{split} \tag{33}$$

That means *GX*(*x*) ≥ *GY*(*x*) and *X* ≤*st Y*.

**Theorem 2.** *Let X* ∼ *GOLE*(*a*1, *b*1, *c*, *φ*) *and Y* ∼ *GOLE*(*a*2, *b*2, *c*, *φ*)*. If a*<sup>1</sup> > *a*<sup>2</sup> *and b*<sup>1</sup> = *b*2*, then X* ≤*lr Y*.

**Proof.** We determine

$$\frac{\mathcal{G}X(\mathbf{x})}{\mathcal{G}Y(\mathbf{x})} = \frac{\left[ \left( a\_1 + (b\_1 - a\_1)F(\mathbf{x})^c \right) \right] \times \exp\left[ -\left( \frac{a\_1 F(\mathbf{x})^c}{1 - F(\mathbf{x})^c} + \frac{b\_1}{2} \left( \frac{F(\mathbf{x})^c}{1 - F(\mathbf{x})^c} \right)^2 \right) \right]}{\left[ \left( a\_2 + (b\_2 - a\_2)F(\mathbf{x})^c \right) \right] \times \exp\left[ -\left( \frac{a\_2 F(\mathbf{x})^c}{1 - F(\mathbf{x})^c} + \frac{b\_2}{2} \left( \frac{F(\mathbf{x})^c}{1 - F(\mathbf{x})^c} \right)^2 \right) \right]}. \tag{34}$$

Thus,

$$\begin{split} \log \left( \frac{\chi\_{X}(x)}{\mathfrak{g}\_{Y}(x)} \right) &= \log \left[ \left( a\_{1} + (b\_{1} - a\_{1})F(x)^{c} \right) \right] - \left( \frac{a\_{1}F(x)^{c}}{1 - F(x)^{c}} + \frac{b\_{1}}{2} \left( \frac{F(x)^{c}}{1 - F(x)^{c}} \right)^{2} \right) \\ &- \log \left[ \left( a\_{2} + (b\_{2} - a\_{2})F(x)^{c} \right) \right] + \left( \frac{a\_{2}F(x)^{c}}{1 - F(x)^{c}} + \frac{b\_{2}}{2} \left( \frac{F(x)^{c}}{1 - F(x)^{c}} \right)^{2} \right) . \end{split} \tag{35}$$

By differentiating the last Equation and after some simplifications, we obtain

$$\frac{d}{dx}\left(\log\left(\frac{q\_X(x)}{g\_Y(x)}\right)\right) = \frac{(a\_2b\_1 - a\_1b\_2)cf(x)F(x)^{\varepsilon-1}}{\left(a\_1 + (b\_1 - a\_1)F(x)^{\varepsilon}\right)\left(a\_2 + (b\_2 - a\_2)F(x)^{\varepsilon}\right)} + \frac{(a\_2 - a\_1)cf(x)F(x)^{\varepsilon-1}}{\left(1 - F(x)^{\varepsilon}\right)^2} + \frac{(b\_2 - b\_1)cf(x)F(x)^{2\varepsilon-1}}{\left(1 - F(x)^{\varepsilon}\right)^3}.\tag{36}$$

Now, if *a*<sup>1</sup> > *a*<sup>2</sup> and *b*<sup>1</sup> = *b*2, then *<sup>d</sup> dx* log *gX*(*x*) *gY*(*x*) <sup>&</sup>lt; 0, and hence *gX*(*x*)/*gY*(*x*) is decreases in *x*. This implies that *X* ≤*lr Y*.
