The Extropy of Concomitants of GeneralizedOrder Statistics from Huang–Kotz–Morgenstern Bivariate Distribution

In this paper, we study the extropy for concomitants of m− generalized order statistics (m− GOSs) from Huang–Kotz–Farlie–Gumbel–Morgenstern (HK-FGM) bivariate distribution. Moreover, the cumulative residual extropy (CREX) and negative cumulative extropy (NCEX) are depicted. Furthermore, the empirical technique in conjunction with the concomitant ofm− GOS is used to investigate the problem of estimating the CREX andNCEX.e concomitants of order statistics and record values are oered as some applications of these ndings.


Introduction
e classic Farlie-Gumbel-Morgenstern (FGM) family of the marginals F X and F Y was treated by Huang and Kotz [1] as a polynomial-type single parameter extension. e distribution function (DF) which they suggested is (1) e corresponding probability density function (PDF) is given by denoted by HK-FGM. e allowable range of the association parameter λ is − (1/p 2 ) ≤ λ ≤ (1/p), and the range for correlation coe cient is − (p + 2) − 2 min(1, p 2 ) ≤ ρ ≤ 3p(p + 2) − 2 . Huang and Kotz [1] revealed that using model (1), the positive correlation between marginal distributions can be enhanced to ≈0.39, while the maximum negative correlation remains − (1/3). Furthermore, when considering model (1) with uniform marginals, the range 0 < p < 1 results in a fast decreasing positive correlation, while the allowable range progressively widens. In addition, at p 1, the highest negative correlation is found. As a result, we will just look at the situation p ≥ 1. It is worth noting that the HK-FGM family's easy analytical form piqued the curiosity of many researchers who wanted to use (and generalize) this model in a variety of elds, e.g., Abd Elgawad et al. [2], Bairamov and Kotz [3], Barakat et al. [4], and Fisher and Klein [5].
Kamps [6] proposed a uni ed model for ordered random variables (RVs) called generalized order statistics (GOSs), which encompass order statistics (OSs), sequential OSs, record values, k− record values, Pfeifer's records, and progressive type II censored OSs as special instances. Many key types of ordered RVs, such as OSs, sequential OSs, lower record values, k− records, and type II censored OSs, are found in the m− GOS subclass of GOSs. Let n ∈ N, m ≥ − 1, k > 0 be parameters such that c i k + (n − i) (m + 1), i 1, 2, . . . , n. e RVs X (r,n,m,k) , r 1, 2, . . . , n, are said to be m− GOSs based on the DF F X (x), if their joint PDF is of the form . e marginal PDF of rth m− GOS, X (r,n,m,k) , 1 ≤ r ≤ n, is given by (cf. [6]) where David [7] proposed the concept of concomitants of OSs for the first time. e concomitants are of interest in problems of selection and prediction. David and Nagaraja [8] provided a comprehensive review of concomitants of OSs. Recently, Barakat et al. [9,10] studied the concomitant of OSs based on the most important extensions of the FGM family. Many authors, including Abd Elgawad et al. [2,11,12], Barakat and Husseiny [13], Alawady et al. [14][15][16], and Tahmasebi et al. [17] have studied the concomitants of GOSs. Assume that (X i , Y i ), i � 1, 2, . . . , n, is a random sample from the given bivariate DF F X,Y (x, y). If the X-variates are arranged in ascending order as X (1,n,m,k) ≤ X (2,n,m,k) ≤ . . . ≤ X (n,n,m,k) , for the X sample, then Y-variates paired with these m− GOSs are called the concomitants of m− GOSs and denoted by Y [r,n,m,k] , r � 1, 2, . . . , n. e PDF of the concomitant of rth m− GOS is given by (see, e.g., Abd Elgawad et al. [2] and Alawady et al. [14]) where f Y|X (y|x) is the conditional PDF of Y given X. As a complement dual of Shannon entropy, Lad et al. [18] proposed extropy as a new measure of uncertainty for an absolutely continuous nonnegative RV X, defined by It is obvious that J(X) ≤ 0. One of the statistical applications of extropy is to score the forecasting distributions using the total log scoring rule (for more details about this measure, see Husseiny et al. [19]). Recently, Almaspoor et al. [20] studied the measures of extropy for concomitants of GOSs in the FGM family. A number of characterizations, as well as extropy lower bounds for OSs and record values, were spotlighted by Qiu [21]. Qiu and Jia [22] looked into residual extropy using OSs, whereas Qiu and Jia [23] looked into extropy estimators used in uniformity testing. e extropy properties of mixed systems were studied by Qiu et al. [24]. A test of uniformity-based extropy was used by Zamanzade and Mahdizadeh [25] to compare ranked set sampling to basic random sampling. Mohamed et al. [26] used fractional and weighted cumulative residual entropy measurements as well as discussed some of its features to test uniformity. Jahanshahi et al. [27] proposed cumulative residual extropy (CREX) as a measure of uncertainty for RV. e CREX is defined as It is always negative. As a result, the negative CREX (NCREX) can be expressed as Newly, Tahmasebi and Toomaj [28] suggested a negative cumulative extropy (NCEX) analogous to (8), defined as where ϕ(u) � (1 − u 2 /2), 0 < u < 1.
In this paper, we aim to investigate the extropy for concomitants of m− GOSs in the HK-FGM family. erefore, the rest of this paper is organized as follows. In Section 2, we begin by calculating the extropy measure of Y [r,n,m,k] in the HK-FGM family. Some results of CREX and NCEX for Y [r,n,m,k] are obtained. In Section 3, the problem of estimating the NCREX and NCEX using the empirical NCREX and NCEX for concomitants of m− GOS is discussed.

Extropy and Some of Its Related Measures in Concomitants of m − GOSs Based on HK-FGM
erefore, the DF and survival function of Y [r,n,m,k] follow If Y [r,n,m,k] is the concomitant of the m− GOS with PDF (11), then the extropy measure is given by where U is a uniformly RV on (0, 1), and J(Y) is the extropy of the RV Y.
Proof. It is easy to check that Δ r,n− 1: p � Δ r,n: p − p(p + 1)β(r + p, n − r + 1) nβ(r, n − r + 1) , (27) and □ Example 1. Assume that X and Y have a joint exponential distribution as HK-FGM (denoted by HK-FGM-ED) By using (15) Example 2. Assume that X and Y have a joint logistic distribution as HK-FGM − λΔ r,n: p Example 3. Assume that X and Y have a joint exponentiated exponential distribution as HK-FGM Remark 2. Assume that (X i , Y i ), i � 1, 2, . . ., is a sequence of bivariate RVs drawn from a continuous DF of a random vector (X, Y). e value of Y that corresponds to the rth upper record R r , r > 1, pertaining to the variate X will be referred to as the rth concomitant of the record value R r and it will be denoted by R [r] . e record values and their corresponding concomitants are applicable in real-world experiments like lifetime studies, sporting events, and other experimental fields.
e record values, record times, and inter-record times were the subject of a statistical discussed by Chandler [30]. Applications of record values and their concomitants have been discussed in Houchens [31], Ahsanullah [32], and Husseiny et al. [19]. e record value is a special case of the m− GOSs with m � − 1 and k � 1. erefore, the PDF and DF for R [r] are obtained as where erefore, the extropy measure for R [r] is obtained as follows:       (2) e value of NCREX is greater than the value of empirical NCREX for p > 1 and 0 < λ ≤ 1.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.