*2.1. Generalized Order Statistics*

The concept of GOS was introduced by Kamps [6], in 1995, who proposed an unifying pattern of various order statistics:

**Definition 1.** *The random variables X*(1, *n*, *m*˜ , *k*)*,. . . , X*(*n*, *n*, *m*˜ , *k*) *are called GOS based on distribution function F with density function f , if their joint density function is given by:*

$$f(\mathbf{x}\_1, \dots, \mathbf{x}\_n) = k \left( \prod\_{j=1}^{n-1} \gamma\_j \right) \left( \prod\_{i=1}^{n-1} (1 - F(\mathbf{x}\_i))^{m\_i} f(\mathbf{x}\_i) \right) (1 - F(\mathbf{x}\_i))^{k-1} f(\mathbf{x}\_n) \tag{1}$$

*on the cone <sup>F</sup>*−1(0) <sup>&</sup>lt; *<sup>x</sup>*<sup>1</sup> <sup>≤</sup> *<sup>x</sup>*<sup>2</sup> ···≤ *xn* <sup>&</sup>lt; *<sup>F</sup>*−1(1) *of* <sup>R</sup>*<sup>n</sup> with parameters <sup>n</sup>* <sup>∈</sup> <sup>N</sup>*, <sup>n</sup>* <sup>≥</sup> <sup>2</sup>*, <sup>k</sup>* <sup>&</sup>gt; <sup>0</sup>*, <sup>m</sup>*˜ = (*m*1,..., *mn*−1)*, <sup>γ</sup><sup>r</sup>* <sup>=</sup> *<sup>k</sup>* <sup>+</sup> *<sup>n</sup>* <sup>−</sup> *<sup>r</sup>* <sup>+</sup> <sup>∑</sup>*n*−<sup>1</sup> *<sup>j</sup>*=*<sup>r</sup> mj* > 0*, for all r* ∈ {1, 2, . . . , *n* − 1}*.*

Some particular cases of GOS are:


As a result of the complex formula of the joint density, finding the marginal distributions of (1) is a difficult task, but in some particular cases, marginal densities can be found. In [6], the marginal densities are determined for *m*<sup>1</sup> = *m*<sup>2</sup> = ... *mn*−<sup>1</sup> = *m*, and in [15] for *γ*<sup>1</sup> = *γj*, 1 ≤ *i* = *j* ≤ *n*. In the following, we will suppose that *m*<sup>1</sup> = *m*<sup>2</sup> = ... *mn*−<sup>1</sup> = *m*, i.e., we are in the m-GOS case where simple order statistics, record values and progressive type II censored order statistics with equi-balanced censoring scheme are included. Now, the density (1) becomes:

$$f(\mathbf{x}\_1, \dots, \mathbf{x}\_n) = k \left( \prod\_{j=1}^{n-1} \gamma\_j \right) \left( \prod\_{i=1}^{n-1} (1 - F(\mathbf{x}\_i))^m f(\mathbf{x}\_i) \right) (1 - F(\mathbf{x}\_i))^{k-1} f(\mathbf{x}\_n) \tag{2}$$

on the cone *<sup>F</sup>*−1(0) <sup>&</sup>lt; *<sup>x</sup>*<sup>1</sup> <sup>≤</sup> *<sup>x</sup>*<sup>2</sup> ··· ≤ *xn* <sup>&</sup>lt; *<sup>F</sup>*−1(1) of <sup>R</sup>*<sup>n</sup>* with parameters *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>, *<sup>n</sup>* <sup>≥</sup> 2, *k* > 0, *m*, *γ<sup>r</sup>* = *k* + (*n* − *r*)(*m* + 1) > 0, for all *r* ∈ {1, 2, . . . , *n* − 1}.

The marginal density function of the r-th GOS, *r* = 1, 2, ... , *n*, in this case, is given by [6]:

$$f\_r(\mathbf{x}) = \frac{c\_{r-1}}{(r-1)!} (1 - F(\mathbf{x}))^{\gamma\_r - 1} f(\mathbf{x}) g\_m^{r-1} (F(\mathbf{x})) \tag{3}$$

where *cr*−<sup>1</sup> <sup>=</sup> <sup>∏</sup>*<sup>r</sup> <sup>j</sup>*=<sup>1</sup> *γ<sup>j</sup>* and for *x* ∈ [0, 1):

$$g\_m(\mathfrak{x}) = \begin{cases} \frac{1}{m+1} (1 - (1 - \mathfrak{x})^{m+1}), & m \neq -1, \\ \log\left(\frac{1}{1 - \mathfrak{x}}\right) & m = -1. \end{cases}$$

**Remark 1.** *For m* = 0 *and k* = 1*, i.e., the case of simple order statistics, we have γ<sup>r</sup>* = *n* − *r* + 1*, cr*−<sup>1</sup> = *γ*1*γ*<sup>2</sup> ... *γ<sup>r</sup>* = *n*(*n* − 1)...(*n* − *r* + 1)*, g*0(*F*(*x*)) = *F*(*x*)*, and* (3) *becomes the wellknown marginal density of the r-th-order statistic.*

*For m* = −1 *and k* = 1*, i.e., the case of record values,* (3) *becomes*

$$f\_r(\mathbf{x}) = \frac{1}{(r-1)!} f(\mathbf{x}) [-\ln(1 - F(\mathbf{x}))]^{n-1},\tag{4}$$

*the marginal density of the r-th record value.*

*For progressive type II censored order statistics with equi-balanced censoring scheme, the form of the marginal density is the same as the form of* (3)*, with m* = *R, R being the removal number.*

### *2.2. Concomitants*

The term concomitant was introduced by David (1973) [1] and has the following definition:

**Definition 2.** *Let* (*X*1,*Y*1),(*X*2,*Y*2), ... ,(*Xn*,*Yn*) *be iid bivariate random variables with cumulative distribution function F*(*x*, *y*)*. Then, the Y variate associated to the r-th-order statistic of X-s, X*(*r*:*n*)*, denoted by Y*[*r*:*n*]*, is the concomitant of X*(*r*:*n*)*.*

A natural use of concomitants is in selection procedures when *k* individuals are chosen on the basis of their *X*-values. Then, the corresponding *Y*-values represent performance on an associated characteristics. In Reliability Theory, the role of the concomitants is emphasized in [3–5].

## *2.3. Concomitants of FGM Family*

The FGM bivariate distribution family has a flexible form and it was studied by Farlie [16], Gumbel [17], Morgenstern [18], and Johnson and Kotz [19].

**Definition 3.** *Let X and Y be two random variables with distribution functions FX and FY, respectively. Additionally, let α be a real number. Then, the FGM family has the distribution function:*

$$F\_{X,Y}(x,y) = F\_X(x)F\_Y(y)[1 + \pi(1 - F\_X(x))(1 - F\_Y(y))].\tag{5}$$

The corresponding probability density function (pdf) of (5) is:

$$f\_{X,Y}(x,y) = f\_X(x)f\_Y(y)[1 + \pi(1 - F\_X(x))(1 - F\_Y(y))],\tag{6}$$

where *fX*(*x*) *fY*(*y*) are the marginals of *fX*,*Y*(*x*, *y*).

The parameter *α* ∈ [−1, 1] is known as the association parameter and the two random variables *X* and *Y* are independent when *α* = 0. For *α* = 0, there is a dependence between the two variables, characterized by the FGM-copula whose properties were studied in [20].

Concomitants of FGM family, related to GOS, started to come into notice with the work of Beg and Ahsanullah in 2008 [21] where the density *g*[*r*,*n*,*m*,*k*] of the concomitant of the r-th GOS is derived:

$$g\_{[r,n,m,k]}(y) = f\_Y(y) + \mathfrak{a}(2F\_Y(y) - 1)f\_Y(y)\mathbb{C}^\*(r,n,m,k),\tag{7}$$

where

$$C^\*(r, n, m, k) = 1 - 2 \frac{\varepsilon\_{r-1}}{(\gamma\_1 + 1)(\gamma\_2 + 1) \dots (\gamma\_r + 1)}$$

is a constant.

$$\text{Remark 2. If } m = 0, k = 1 \text{, then } \mathbb{C}^\*(r, n, 0, 1) = -(n - 2r + 1)/(n + 1) \text{ and}$$

$$g\_{[r,n,0,1]}(y) = f\_Y(y) - \mathfrak{a}\frac{n-2r+1}{n+1}(2F\_Y(y)-1)f\_Y(y) \tag{8}$$

*is the density of the concomitant of r-th-order statistic from the FGM family. If m* <sup>=</sup> <sup>−</sup>1*, k* <sup>=</sup> <sup>1</sup>*, then C*∗(*r*, *<sup>n</sup>*, <sup>−</sup>1, 1) = <sup>1</sup> <sup>−</sup> <sup>2</sup>1−*<sup>r</sup> and*

$$g\_{\left[r,n,-1,1\right]}(y) = f\_Y(y) - a(2^{1-r}-1)(2F\_Y(y)-1)f\_Y(y). \tag{9}$$

*If we are in the case of progressive type II censoring order statistics with equi-balanced censoring scheme, the density of the concomitant of r-th-order statistic from the FGM family is* (7) *with m* = *R, the removal number.*

The cumulative distribution function and the survival function of the concomitant of r-th-order statistic can also be computed:

$$G\_{[r,n,m,k]}(y) = f\_Y(y) + a(1 - F\_Y(y)) f\_Y(y) \mathbb{C}^\*(r, n, m, k),\tag{10}$$

$$\overline{G}\_{[r,n,m,k]}(y) = 1 - f\_Y(y) - a(1 - F\_Y(y))f\_Y(y)\mathbb{C}^\*(r,n,m,k). \tag{11}$$

In the following, in order to make it easier to read computations, we make the notations: *Y*∗ [*r*] <sup>=</sup> *<sup>Y</sup>*[*r*,*n*,*m*,*k*], *<sup>g</sup>*[*r*] <sup>=</sup> *<sup>g</sup>*[*r*,*n*,*m*,*k*], *<sup>G</sup>*[*r*] <sup>=</sup> *<sup>G</sup>*[*r*,*n*,*m*,*k*], *<sup>G</sup>*¯ [*r*] = *G*¯ [*r*,*n*,*m*,*k*], *C*<sup>∗</sup> *<sup>r</sup>* = *C*∗(*r*, *n*, *m*, *k*).

#### **3. Information Measures for the Concomitants from the FGM Family, Existing Results**

In this section, we will recall some definitions and results for the information measures of the concomitants of GOS from the FGM family.
