*4.3. Order Statistics*

For a random sample of *X*1, ... , *Xn* taken from the OGE2Fr distribution with *Xi*:*n* as the *i*th order statistic. For *i* = 1, 2, 3, ... , *n*, the pdf corresponding to *Xi*:*n* can be expressed as

$$\begin{aligned} f\_{i:n}(\mathbf{x}) &= \frac{1}{\beta(i, n-i+1)} f(\mathbf{x}) F(\mathbf{x})^{i-1} \left\{ 1 - F(\mathbf{x}) \right\}^{n-i} \\ &= \frac{1}{\beta(i, n-i+1)} \sum\_{j=0}^{n-i} (-1)^j \binom{n-i}{j} f(\mathbf{x}) F(\mathbf{x})^{j+i-1} \end{aligned}$$

where *f*(*x*) and *<sup>F</sup>*(*x*) are the pdf and cdf of OGE2Fr distribution, respectively. Inserting Equations (17) and (18), and using the result defined in Section 3, we have

$$f\_{i:n}(\mathbf{x}; m\, a, b) = \sum\_{j=0}^{n-i} \eta\_j \, m \, b \, a^b \, \mathbf{x}^{-b-1} \mathbf{e}^{-m(a/x)^b},\tag{26}$$

where

$$\eta\_{\vec{\beta}} = \frac{(-1)^{\vec{j}}}{\beta(\vec{i}, n-\vec{i}+1)} \binom{n-\vec{i}}{\vec{j}} \sum\_{m=0}^{\infty} \xi\_{m}^{\ast \ast}$$

and

$$\zeta\_{m}^{\chi\_{m}^{\bullet}} = (-1)^{m} \sum\_{i,k=0}^{\infty} \sum\_{l=0}^{i} \frac{(-1)^{i+k+l} (k+1)^{i} \Gamma\{\beta(i+j)\}}{i! k! \Gamma\{\beta(i+j) - k\}} \binom{i}{l} \binom{-a(i+l+1)-1}{m} \dots$$

*fi*:*n*(*<sup>x</sup>*; *m a*, *b*) is the probability density function of the OGE2Fr distribution with parameters *m a* and *b*.
