Ordered Variables and Their Concomitants under Extropy via COVID-19 Data Application

Extropy, as a complementary dual of entropy, has been discussed in many works of literature, where it is declared for other measures as an extension of extropy. In this article, we obtain the extropy of generalized order statistics via its dual and give some examples from well-known distributions. Furthermore, we study the residual and past extropy for such models. On the other hand, based on Farlie–Gumbel–Morgenstern distribution, we consider the residual extropy of concomitants of m-generalized order statistics and present this measure with some additional features. In addition, we provide the upper bound and stochastic orders of it. Finally, nonparametric estimation of the residual extropy of concomitants ofm-generalized order statistics is included using simulated and real data connected with COVID-19 virus.


Introduction
Shannon [1] introduced a well-known vintage measure of uncertainty called Shannon entropy.
is information theoretic entropy manipulates in diverse fields such as financial analysis, computer science, and medical research. e extropy proposed by Lad et al. [2] is an accomplishment to notions of information based on entropy. ey exhibited that entropy has a complementary dual function known as "extropy." In the view of extropy in discrete density, the extropy measure − N i�1 (1 − θ i )log(1 − θ i ) is neatly closer to (− 1/2) N i�1 θ 2 i when the range of possibilities increases (as a result of larger N). erefore, to realize extropy for a continuous density, the extropy of a nonnegative continuous random variable (r.v.) X, with probability density function (PDF) f(x) is defined as e extropy measure has been developed for ordered variables. Qiu [3] was the first to apply extropy for order statistics and record values and present several of their properties. After that, the researchers manifested to present extension measures of extropy. Qiu and Jia [4] investigated the connotation of residual extropy of a nonnegative r.v. as Qiu et al. [5] presented a mixed systems lifetime via extropy and obtained some features and bounds of it. Recently, Jose and Sathar [6,7] exploited the residual and past extropy of k-records, respectively, emerging from any continuous distribution. For extra studies on extropy, see Qiu and Jia [8], Yang et al. [9], Noughabi and Jarrahiferiz [10], Raqab and Qiu [11], and Lad et al. [12].
Krishnan et al. [13] presented the past extropy, for a fixed t > 0, for past lifetime of r.v. X t � [t − X|X ≤ t] as follows: Jahanshahi et al. [14] proposed cumulative residual extropy (CREX). For a nonnegative r.v. X with an absolutely continuous survival function F, the CREX is given by In analogy with Jahanshahi et al. [14], Abdul Sathar and Dhanya [15] introduced the CREX and refer to it as survival extropy. Moreover, they conduct the dynamic survival extropy as It is easy to observe that extropy and its related measures are constantly negative. e connotation of generalized order statistics (gos) that contains all forms of ordered random observations was first proposed by Kamps [16]. Let k ≥ 1, n ∈ N, m � (m 1 , . . . , m n− 1 ) ∈ R n− 1 be parameters such that M r � n− 1 j�r m j , c r � k + n − r + M r ≥ 1 for all 1 ≤ r ≤ n − 1. For a subclass of gos (called m − gos), where m 1 � m 2 � · · · � m n− 1 � m, the PDF of the rth m − gos, X (r;n,k,m) , can be written as f (r;n,k,m) (x) � c r− 1 (r − 1)! (1 − F(x)) c r − 1 f(x)g r− 1 m (F(x)), (6) where c r− 1 � r j�1 c j , c r � k + (n − r)(m + 1), for 0 < z < 1, We will write g m (z) Based on descending ordered r.v.'s, Pawlas and Szynal [17] and Burkschat et al. [18] presented the dual generalized order statistics (dgos). By the same manner and parameters in m − gos, when m 1 � m 2 � · · · � m n− 1 � m, the PDF of m − dgos X d(r;n,k,m) is defined by e concept of concomitants of ordinary order statistic was derived by David et al. [19]. Let (X i , Y i ), i � 1, 2, . . . , n, be n pairs of independent r.v.'s drawn from some bivariate distributions with cumulative distribution function (CDF) F(x, y). Let X (r;n) be the rth order statistic, then the r.v. Y concerned with X (r;n) is called the concomitant of rth order statistics and is specified by Y [r;n] . e Farlie-Gumbel-Morgenstern (FGM) family is an extremely supple class of bivariate family; it was primarily derived by Morgenstern [20], which is set by CDF and PDF, respectively, as follows: where F X (x), F Y (y) and f X (x), f Y (y) are the marginal CDF's and PDF's of X and Y, respectively, − 1 ≤ α ≤ 1. If the dependent parameter α � 0, then X and Y are not dependent. Beg and Ahsanullah [21] introduced the PDF of the concomitant of m − gos Y [r;n,k,m] , 1 ≤ r ≤ n, under the FGM family as follows: where roughout this paper, we propose the extropy of m − gos and m − dgos and study those models for the related measures of extropy. In the second part of the paper, we deal with the concomitants of m − gos of FGM family to extract the residual extropy and give some of its properties. e paper is organized as follows: Section 2 contains extropy of m − gos and m − dgos obtained from uniform distribution. Moreover, we obtain them for any distribution in terms of the obtained extropy from uniform distribution. Meanwhile, we produce some examples of some well-known distributions. Furthermore, we obtain the lower bound of the extropy of m − gos in terms of the mode. In addition, the residual and past extropy of m − gos and m − dgos is considered in Section 3. In Section 4, we derive the residual extropy of concomitants of m − gos of FGM family and discuss its relation with the stop-loss transform and Gini index. Besides, we consider this model in terms of its upper bound and produce some examples on it. Finally, in Section 5, real-life data connected with the COVID-19 virus is applied for the nonparametric estimation of residual extropy of concomitants of order statistics under the FGM family.

Extropy of m-Generalized Order Statistics and Its Dual
In this section, we discuss the extropy of m − gos and m − dgos for uniform distribution and for any distribution, which depends on beta function and its generalized first kind.
From the previous theorem, we show that the extropy of m − gos is the product of extropy of m − gos emerging from U(0, 1) distribution and expectation of the first kind generalized beta distributed r.v. Now, we will give some special cases on eorem 2 by the following examples. □ Example 1. Suppose the nonnegative continuous r.v. X arising from exponential distribution, denoted by EXP(λ), with CDF erefore, us, By the same manner in (17), we can reduce (25) as Example 2. Suppose the nonnegative continuous r.v. X arising from Pareto distribution with CDF erefore, us, ence, In the next corollary, the lower bound for the extropy of m − gos will be obtained in terms of the extropy of m − gos emerging from U(0, 1) and the mode of the distribution.  (1) and (8) and eorem 1, we can obviously see that the extropy of rth m − gos, U (r;n,k,m) , is the same as the extropy of the rth m − dgos U d(r;n,k,m) . On the other hand, we can obtain the extropy of rth m − dgos, X d(r;n,k,m) , 1 ≤ r ≤ n, for any distribution from the following theorem. (1), (8), and (13), the extropy of the rth m − dgos, X d(r;n,k,m) , 1 ≤ r ≤ n, is given by (19).

Theorem 3. Let X be a continuous r.v. that is nonnegative with CDF F(x) distribution. en, from
Proof. From (1) and (8), we have (33) where the extropy of rth m − dgos J(U d(r;n,k,m) ) � J (U (r;n,k,m) ) obtained in (13), which proves the theorem. □

Residual Extropy of Concomitants of m-Generalized Order Statistics
In this section, we will discuss the residual extropy of concomitants of m − gos under the FGM family. From equation (9), the conditional CDF of Y given X � x is given by Under the FGM family with conditional CDF given by equation (42), Mohie El-Din et al. [23] presented the CDF of the concomitant of m − gos Y [r;n,k,m] , 1 ≤ r ≤ n, as follows: where f (r;n,k,m) (x) is the PDF of m − gos X (r;n,k,m) . erefore, From (4) and (44), the residual extropy of concomitants of m − gos is given by Furthermore, we can write (45) in terms of the moments as follows: where μ n: n � ∞ 0 yf y n: n (y)dy and y n: n is the nth order statistic of a random sample of size n of the Y variate, μ � E(Y).

Stop-Loss Transform and Gini Coefficient.
In this section, we will present ζ [r;n,k,m] (Y) related to stop-loss transform and Gini index.
Definition 2. Suppose X and Y are independent r.v.'s and have the same distribution as X. en, the Gini index or Gini coefficient is given by (see Wang [24] for more details).

Remark 1.
From Jahanshahi et al. [14], based on (4) and Definitions 1 and 2, we have where m F (Y) � (Z F (Y)/F Y (y)) is the mean residual life function. (4). en, the cumulative residual extropy of concomitants of m − gos, Y (r;n,k,m) , can be expressed as

Stochastic Orders
Definition 3. Y 2 is known to be smaller than Y 1 in the usual stochastic order, denoted by Y 2 ≤ st Y 1 if and only if F Y 2 (t) ≤ F Y 1 (t), for all t ∈ R. For more details, see Shaked and Shanthikumar [28].

Theorem 12.
Let Y 1 and Y 2 be two nonnegative continuous r.v.'s with CDF's F Y 1 (·) and F Y 2 (·) and finite mean E(Y 1 ) and E(Y 2 ), respectively. If Y 2 ≤ st Y 1 , then we have the following: (1) From Remark 2, under the conditions on the parameters in (54), we get (2) From Remark 2, under the conditions on the parameters in (55), we get Proof. First, from eorem 7 of Jahanshahi et al. [14], if . erefore, the proof of (1) is e proof of (2) is Now, we will give an application of the last theorem as follows.

Nonparametric Estimation
In this section, we obtain a nonparametric estimation of the residual extropy of concomitants of m − gos under the FGM family by the empirical data. Let Y 1 , . . . , Y n be a random sample from a population with CDF F and its empirical estimator F n . From (45), the empirical residual extropy of concomitants of m − gos is given by where F n (y) � 1 − F n (y), F n (y) is the empirical CDF, and Y (1) ≤ Y (2) ≤ · · · ≤ Y (n) are the associated order statistics of the random sample.
In the following examples, we apply the proposed methods to explain the performance of the empirical and kernel estimators.
Example 7. Let X 1 , . . . , X n be a random sample of uniform distribution U(0, 1). According to Pyke [30], the sample spacing W j+1 follows the beta distribution Beta(1, n). Hence, from (70)  (73) Example 8. Let X 1 , . . . , X n be a random sample of Exp(λ). According to Pyke [30], the sample spacing W j+1 follows Exp(λ(n − j)). Hence, from (70)  (n − j) 2 1 + αT * (r; n, k, m)F n y j 4 . (74) Based on order statistics (k � 1, m � 0), Table 1 presents the mean and variance of ζ 1[r;n,1,0] and ζ 2[r;n,1,0] from U(0, 1) and EXP(1), respectively, by using different values of sample size (n � 10, 30, 50, 70, 100). In Table 1, for fixed n and r increases, we conclude that the mean decreases and the variance increases. We use Kolmogorov-Smirnov (K-S) test to check the fitting of the data for EXP (1), which implies that the K-S statistic is 0.076282 with p value 0.9674. us, it is admitted to fit the data by EXP(1); furthermore, see Figure 1. Based on EXP(1), Figures 2 and 3 present the real-life and simulated data, respectively. erefore, we can conclude that by decreasing α and increasing r, the empirical estimators approach the theoretical value and vice versa.  Finally, we considered the problem of estimating ζ [r;n,k,m] (Y) by proposing two different empirical estimators of CDF. We concluded that the proposed estimators are affected by sample size n, r, and αand generally the first empirical estimator is more accurate than the second estimator.

Data Availability
All the data sets are provided within the main body of the paper.

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