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

Authors: Mohamed S. Mohamed; Alanazi Talal Abdulrahman; Zahra Almaspoor; M. Yusuf
Journal: Complexity
 (2021)

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## Abstract

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 ofm-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 of m-generalized order statistics is included using simulated and real data connected with COVID-19 virus.


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## Body

## 1. Introduction

Shannon [1] introduced a well-known vintage measure of uncertainty called Shannon entropy. This information theoretic entropy manipulates in diverse fields such as financial analysis, computer science, and medical research. The extropy proposed by Lad et al. [2] is an accomplishment to notions of information based on entropy. They exhibited that entropy has a complementary dual function known as “extropy.” In the view of extropy in discrete density, the extropy measure −∑i=1N1−θilog1−θi is neatly closer to −1/2∑i=1Nθi2 when the range of possibilities increases (as a result of larger N). Therefore, to realize extropy for a continuous density, the extropy of a nonnegative continuous random variable (r.v.) X, with probability density function (PDF) fx is defined as(1)JX=−12∫0∞f2xdx.The 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(2)JtX=JRX;t=−12F¯2t∫t∞f2xdx,t≥0.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. Xt=t−X|X≤t as follows:(3)JtX=JPX;t=−12F2t∫0tf2xdx.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(4)ζX=−12∫0∞F¯2xdx.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(5)ζX;t=−12F¯2t∫t∞F¯2xdx,t≥0.It is easy to observe that extropy and its related measures are constantly negative.The 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˜=m1,…,mn−1∈ℝn−1 be parameters such that Mr=∑j=rn−1mj, γr=k+n−r+Mr≥1 for all 1≤r≤n−1. For a subclass of gos (called m−gos), where m1=m2=⋯=mn−1=m, the PDF of the rthm−gos, Xr;n,k,m, can be written as(6)fr;n,k,mx=cr−1r−1!1−Fxγr−1fxgmr−1Fx,where cr−1=∏j=1rγj, γr=k+n−rm+1, for 0<z<1,(7)gmz=1−1−zm+1m+1,m≠−1,−ln1−z,m=−1.We will writegmz=1−1−zm+1/m+1, 0<z<1 and for all m, since limm⟶−11−1−zm+1/m+1=−ln1−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 m1=m2=⋯=mn−1=m, the PDF of m−dgosXdr;n,k,m is defined by(8)fdr;n,k,mx=cr−1r−1!1m+1r−1Fxγr−11−Fm+1xr−1fx.The concept of concomitants of ordinary order statistic was derived by David et al. [19]. Let Xi,Yi, i=1,2,…,n, be n pairs of independent r.v.’s drawn from some bivariate distributions with cumulative distribution function (CDF) Fx,y. Let Xr;n be the rth order statistic, then the r.v. Y concerned with Xr;n is called the concomitant of rth order statistics and is specified by Yr;n.The 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:(9)FX,Yx,y=FXxFYy1+α1−FXx1−FYy,(10)fX,Yx,y=fXxfYy1+α2FXx−12FYy−1,where FXx, FYy and fXx, fYy 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−gosYr;n,k,m, 1≤r≤n, under the FGM family as follows:(11)gr;n,k,my=fYy1+αT∗r;n,k,m2FYy−1,where(12)T∗r;n,k,m=1−2∏j=1rγj∏i=1rγi+1,with parameters n∈ℕ, k≥1, m∈ℝ, such that γr=k+n−rm+1≥1, for all 1≤r≤n.Throughout this paper, we propose the extropy ofm−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. The 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.


## 2. Extropy ofm-Generalized Order Statistics and Its Dual

In this section, we discuss the extropy ofm−gos and m−dgos for uniform distribution and for any distribution, which depends on beta function and its generalized first kind.Based on uniform distributionU0,1, the extropy of the rthm−gos, Xr;n,k,m, 1≤r≤n, can be obtained by the following theorem.Theorem 1.
Suppose the nonnegative continuous r.v.X be emerging from U0,1 distribution. Let Ur;n,k,m be its rthm−gos; then, from (1) and (6), the extropy of Ur;n,k,m is given by(13)JUr;n,k,m=−cr−122r−1!21m+12r−1B2r−1,2γr−1m+1=−cr−122r−1!22r−2!∏i=1rγ1+γi−1∏i=1r−1γr+γ1+i−1.Proof.
From (1) and (6), we have(14)JUr;n,k,m=−12∫01fr;n,k,m2udu=Ar;n,k,m∫011−u2γr−21−1−um+12r−2du,where Ar;n,k,m=−1/2cr−12/r−1!21/m+12r−2. Using the transformation z=1−u, thus(15)JUr;n,k,m=Ar;n,k,m∫01z2γr−21−zm+12r−2dz=Ar;n,k,m1m+1B2r−1,2γr−1m+1,where Bx,y/z=z∫01ty−11−tzx−1dt=ΓxΓy/z/Γx+y/z is the beta function. Therefore, we can reduce the beta function in (15) as follows:(16)B2r−1,2γr−1m+1=Γ2r−1Γ2γr−1/m+1Γ2r−1+2γr−1/m+1=2r−2!Γ2γr−1/m+12r+2γr−1/m+1−22r+2γr−1/m+1−3,…,2γr−1/m+1+r−1︸r−times×12γr−1/m+1+r−22γr−1/m+1+r−3,…,2γr−1/m+1Γ2γr−1/m+1︸r−1−times.
Furthermore,(17)1m+12r−1B2r−1,2γr−1m+1=1m+1r1m+1r−1B2r−1,2γr−1m+1=1m+1r1m+1r−12r−2!2r+2γr−1/m+1−2,…,2γr−1/m+1+r−1︸r−times×12γr−1/m+1+r−22γr−1/m+1+r−3,…,2γr−1/m+1︸r−1−times=2r−2!∏i=1rγ1+γi−1∏i=1r−1γr+γ1+i−1,and the result is taken after.
In the following theorem, we derive the extropy ofm−gos, Xr;n,k,m, from any continuous distribution based on extropy of m−gos emerging from U0,1.Theorem 2.
LetX be a continuous r.v. that is nonnegative with CDF Fx. Then, from (1), (6), and (13), the extropy of the rthm−gos, Xr;n,k,m, 1≤r≤n, is given by(18)JXr;n,k,m=JUr;n,k,mEfF−11−Vr,where Vr has generalized beta of first kind distribution (i.e., Vr∼GB1x;m+1,2γr−1/m+1,2r−1) and GB1x;a,p,q has PDF:(19)fx;a,p,q=axap−11−xaq−1Bp,q,0<x<1,a,p,q>0.Proof.
From (1) and (6), we have(20)JXr;n,k,m=−12∫0∞fr;n,k,m2xdx=Ar;n,k,m∫0∞1−Fx2γr−21−1−Fxm+12r−2f2xdx.
Puttingu=1−Fx, thus(21)JXr;n,k,m=Ar;n,k,m∫01u2γr−21−um+12r−2fF−11−udu=Ar;n,k,m1m+1B2r−1,2γr−1/m+1×m+1B2r−1,2γr−1/m+1∫01u2γr−21−um+12r−2fF−11−udu=JUr;n,k,mEfF−11−Vr,which proves the theorem.
From the previous theorem, we show that the extropy ofm−gos is the product of extropy of m−gos emerging from U0,1 distribution and expectation of the first kind generalized beta distributed r.v.
Now, we will give some special cases on Theorem2 by the following examples.Example 1.
Suppose the nonnegative continuous r.v.X arising from exponential distribution, denoted by EXPλ, with CDF(22)Fx=1−e−λx,λ>0,x>0,F−1x=−1λln1−x.
Therefore,(23)fF−11−u=λu.
Thus,(24)EfF−11−Vr=λB2r−1,2γr−1/m+1B2r−1,2γrm+1.
Thence,(25)JXr;n,k,m=−λcr−122r−1!21m+12r−1B2r−1,2γrm+1.
By the same manner in (17), we can reduce (25) as(26)JXr;n,k,m=−λcr−122r−1!22r−2!∏i=1rγ1+γi∏i=1r−1γr+γ1+i.Example 2.
Suppose the nonnegative continuous r.v.X arising from Pareto distribution with CDF(27)Fx=1−xσ−α,σ>0,x>σ,F−1x=σ1−x−1/α,
Therefore,(28)fF−11−u=ασu1+1/α.
Thus,(29)EfF−11−Vr=ασB2r−1,2γr−1/m+1B2r−1,2γr+1/αm+1.
Thence,(30)JXr;n,k,m=−αcr−122σr−1!21m+12r−1B2r−1,2γr+1/αm+1.In the next corollary, the lower bound for the extropy ofm−gos will be obtained in terms of the extropy of m−gos emerging from U0,1 and the mode of the distribution.Corollary 1.
Suppose thatM is the mode of the r.v. X such that M=fm which exists; therefore,(31)JXr;n,k,m≥JUr;n,k,mM.


### 2.1. Extropy of Dualm-Generalized Order Statistics

Based onU0,1 distribution, from (1) and (8) and Theorem 1, we can obviously see that the extropy of rthm−gos, Ur;n,k,m, is the same as the extropy of the rthm−dgosUdr;n,k,m. On the other hand, we can obtain the extropy of rthm−dgos, Xdr;n,k,m, 1≤r≤n, for any distribution from the following theorem.Theorem 3.
LetX be a continuous r.v. that is nonnegative with CDF Fx distribution. Then, from (1), (8), and (13), the extropy of the rthm−dgos, Xdr;n,k,m, 1≤r≤n, is given by(32)JXdr;n,k,m=JUdr;n,k,mEfF−1Vr,where Vr∼GB1x;m+1,2γr−1/m+1,2r−1 with PDF defined in (19).Proof.
From (1) and (8), we have(33)JXdr;n,k,m=−12∫0∞fdr;n,k,m2xdx=Ar;n,k,m∫0∞Fx2γr−21−Fxm+12r−2f2xdx.
Puttingu=Fx, thus(34)JXdr;n,k,m=Ar;n,k,m1m+1B2r−1,2γr−1m+1×m+1B2r−1,2γr−1/m+1∫01u2γr−21−um+12r−2fF−1udu=JUdr;n,k,mEfF−1Vr,where the extropy of rthm−dgosJUdr;n,k,m=JUr;n,k,m obtained in (13), which proves the theorem.


## 2.1. Extropy of Dualm-Generalized Order Statistics

Based onU0,1 distribution, from (1) and (8) and Theorem 1, we can obviously see that the extropy of rthm−gos, Ur;n,k,m, is the same as the extropy of the rthm−dgosUdr;n,k,m. On the other hand, we can obtain the extropy of rthm−dgos, Xdr;n,k,m, 1≤r≤n, for any distribution from the following theorem.Theorem 3.
LetX be a continuous r.v. that is nonnegative with CDF Fx distribution. Then, from (1), (8), and (13), the extropy of the rthm−dgos, Xdr;n,k,m, 1≤r≤n, is given by(32)JXdr;n,k,m=JUdr;n,k,mEfF−1Vr,where Vr∼GB1x;m+1,2γr−1/m+1,2r−1 with PDF defined in (19).Proof.
From (1) and (8), we have(33)JXdr;n,k,m=−12∫0∞fdr;n,k,m2xdx=Ar;n,k,m∫0∞Fx2γr−21−Fxm+12r−2f2xdx.
Puttingu=Fx, thus(34)JXdr;n,k,m=Ar;n,k,m1m+1B2r−1,2γr−1m+1×m+1B2r−1,2γr−1/m+1∫01u2γr−21−um+12r−2fF−1udu=JUdr;n,k,mEfF−1Vr,where the extropy of rthm−dgosJUdr;n,k,m=JUr;n,k,m obtained in (13), which proves the theorem.


## 3. Residual and Past Extropy ofm-Generalized Order Statistics and Its Dual

Under the condition of parameters in (6), Kamps [16] obtained the CDF of m−gos, Xr;n,k,m, 1≤r≤n, as follows:(35)Fr;n,k,mx=1−cr−11−Fxγr∑j=0r−11j!cr−j−11m+11−1−Fxm+1j,where cr−j−1=∏i=1r−jγi. Furthermore, Shahbaz et al. [22] derived the CDF of both m−gos and m−dgos, Xr;n,k,m, Xdr;n,k,m, 1≤r≤n, respectively, in terms of incomplete beta function ratio as follows:(36)Fr;n,k,mx=IβFxr,γrm+1,(37)Fdr;n,k,mx=Iβ∗Fxr,γrm+1,where Ixa,b=∫0xta−11−tb−1dt is incomplete beta function ratio, βFx=1−1−Fxm+1, β∗Fx=Fxm+1.In the next theorems, we obtain the residual and past extropy ofm−gos and m−dgos emerging from any distribution.Theorem 4.
LetX be a continuous r.v. that is nonnegative with CDF Fx distribution. Then, from (2), (6), and (36), the residual extropy of rthm−gos, Xr;n,k,m, 1≤r≤n, is given by(38)JXr;n,k,m=1F¯r;n,k,m2xJUr;n,k,mEfF−11−Vr|Vr<1−Ft,where Vr∼GB1x;m+1,2γr−1/m+1,2r−1 with PDF defined in (19) and F¯r;n,k,mx=1−Fr;n,k,mx defined in (36).Theorem 5.
LetX be a continuous r.v. that is nonnegative with CDF Fx distribution. Then, from (3), (6), and (36), the past extropy of the rthm−gos, Xr;n,k,m, 1≤r≤n, is given by(39)JXr;n,k,m=1Fr;n,k,m2xJUr;n,k,mEfF−11−Vr|Vr>1−Ft,where Vr∼GB1x;m+1,2γr−1/m+1,2r−1 with PDF defined in (19) and Fr;n,k,mx defined in (36).Theorem 6.
LetX be a continuous r.v. that is nonnegative with CDF Fx distribution. Then, from (2), (8), and (37), the residual extropy of the rthm−dgos, Xdr;n,k,m, 1≤r≤n, is given by(40)JXdr;n,k,m=1F¯dr;n,k,m2xJUdr;n,k,mEfF−1Vr|Vr<1−Ft,where Vr∼GB1x;m+1,2γr−1/m+1,2r−1 with PDF defined in (19) and F¯dr;n,k,mx=1−Fdr;n,k,mx defined in (37).Theorem 7.
LetX be a continuous r.v. that is nonnegative with CDF Fx distribution. Then, from (3), (8), and (37), the past extropy of the rthm−dgos, Xdr;n,k,m, 1≤r≤n, is given by(41)JXdr;n,k,m=1Fdr;n,k,m2xJUdr;n,k,mEfF−1Vr|Vr>1−Ft,where Vr∼GB1x;m+1,2γr−1/m+1,2r−1 with PDF defined in (19) and Fdr;n,k,mx defined in (37).


## 4. Residual Extropy of Concomitants ofm-Generalized Order Statistics

In this section, we will discuss the residual extropy of concomitants ofm−gos under the FGM family. From equation (9), the conditional CDF of Y given X=x is given by(42)FY|Xy|x=FYy1+α1−2FXx1−FYy.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−gosYr;n,k,m, 1≤r≤n, as follows:(43)Gr;n,k,my=FYy1−αT∗r;n,k,m1−FYy,where fr;n,k,mx is the PDF of m−gosXr;n,k,m. Therefore,(44)G¯r;n,k,my=1−Gr;n,k,my=1−FYy1−αT∗r;n,k,m1−FYy=F¯Yy1+αT∗r;n,k,mFYy.From (4) and (44), the residual extropy of concomitants of m−gos is given by(45)ζr;n,k,mY=−12∫0∞G¯r;n,k,my2dy=−12∫0∞F¯Y2y+2αT∗r;n,k,mF¯Y2yFYy+αT∗r;n,k,m2F¯Y2yFY2ydy=ζY+αT∗r;n,k,m2−4αT∗r;n,k,mEYFYy+3αT∗r;n,k,m−αT∗r;n,k,m2EYFY2y+2αT∗r;n,k,m2EYFY3y+αT∗r;n,k,mEY.Furthermore, we can write (45) in terms of the moments as follows:(46)ζr;n,k,mY=ζY+12αT∗r;n,k,m2−4αT∗r;n,k,mμ2:2+αT∗r;n,k,m−αT∗r;n,k,m2μ3:3+12αT∗r;n,k,m2μ4:4+αT∗r;n,k,mμ,where μn:n=∫0∞yfyn:nydy and yn:n is the nth order statistic of a random sample of size n of the Y variate, μ=EY.


### 4.1. Stop-Loss Transform and Gini Coefficient

In this section, we will presentζr;n,k,mY related to stop-loss transform and Gini index.Definition 1.
The stop-loss transformZFt of the nonnegative r.v. Y is defined as(47)ZFt=EmaxY−t,0=∫t∞F¯Yydy.Definition 2.
SupposeX and Y are independent r.v.’s and have the same distribution as X. Then, the Gini index or Gini coefficient is given by(48)Gindex=EX−YEX+Y=1−∫0∞F¯X2xdxEX,(see Wang [24] for more details).Remark 1.
From Jahanshahi et al. [14], based on (4) and Definitions 1 and 2, we have(49)ζY=−12EY−EZFY=−12EY−EF¯YymFY=EY2Gindex−1,where mFY=ZFY/F¯Yy is the mean residual life function.Theorem 8.
LetY be a continuous r.v. that is nonnegative with CREX ζX defined in (4). Then, the cumulative residual extropy of concomitants of m−gos, Yr;n,k,m, can be expressed as(50)ζr;n,k,mY=12EZFY+Tr;n,k,m∗=12EF¯YymFY+Tr;n,k,m∗=12EYGindex+Tr;n,k,m∗,where(51)Tr;n,k,m∗=αT∗r;n,k,m2−4αT∗r;n,k,mEYFYy+3αT∗r;n,k,m−αT∗r;n,k,m2EYFY2y+2αT∗r;n,k,m2EYFY3y+αT∗r;n,k,mEY.Proof.
The proof directly follows from (45) and Remark 1.


### 4.2. Upper Bound ofζr;n,k,mY

In another view of the cumulative residual extropy of concomitants ofm−gos, ζr;n,k,mY, we present it depending on its upper bound.Theorem 9.
LetY be a continuous r.v. that is nonnegative with survival function F¯Yy. Based on FGM family with CDF given by (9), utilizing (4), the residual extropy of the rth concomitants m−gos, Yr;n,k,m, 1≤r≤n, can be expressed by(52)ζr;n,k,mY≤ζy+αT∗r;n,k,m3EYFY2y+EY−4EYFYy.Proof.
Using Bernoulli’s inequality, from (44), we have(53)ζr;n,k,mY=−12∫0∞G¯r;n,k,my2dy=−12∫0∞F¯Y2y1+αT∗r;n,k,mFYy2dy≤−12∫0∞F¯Y2y1+2αT∗r;n,k,mFYydy=ζy+αT∗r;n,k,m3EYFY2y+EY−4EYFYy,where T∗r;n,k,m is defined in (12).
From the previous theorem, we can conclude the following remark.Remark 2.
Since−1<T∗r;n,k,m<1, then the residual extropy of the concomitant of rthm−gos given in (53) can be considered in following cases:(1)
IfT∗r;n,k,m=0, then ζr;n,k,mY=ζy.(2)
If0<α<1 and 0<T∗r;n,k,m<1 or −1<α<0 and −1<T∗r;n,k,m<0, then(54)ζr;n,k,mY<ζy+αT∗r;n,k,m3EYFY2y+EY.(3)
If0<α<1 and −1<T∗r;n,k,m<0 or −1<α<0 and 0<T∗r;n,k,m<1, then(55)ζr;n,k,mY<ζy−4αT∗r;n,k,mEYFYy.Theorem 10.
LetY be a continuous r.v. that is nonnegative with survival function F¯Yy. Then, we have the following:(1)
From Remark2, under the conditions on the parameters in (54), we get(56)ζr;n,k,mY<12ξY+Fr;n,k,m∗≤EY24EY+Fr;n,k,m∗.(2)
From Remark2, under the conditions on the parameters in (55), we get(57)ζr;n,k,mY<12ξY+Rr;n,k,m∗≤EY24EY+Rr;n,k,m∗,where ξY=−∫0∞F¯YylogF¯Yydy is the cumulative residual entropy presented by Rao et al. [25]:(58)Fr;n,k,m∗=3αT∗r;n,k,mEYFY2y+EYαT∗r;n,k,m−12,(59)Rr;n,k,m∗=−4αT∗r;n,k,mEYFYy−12EY.Proof.
The proof directly follows from inequality−logx≥1−x,x>0 and ξY≤EY2/2EY; see Rao et al. [25].
In the next remark, we will give an application of the previous results, taking the order statistics (withk=1 and m=0) as a special case under the FGM family.Remark 3.
For order statistic (m=0 and k=1), we have T∗r;n,1,0=−n−2r+1/n+1. Moreover, by controlling the values of n and r, we can observe the following:(1)
T∗r;n,1,0=0, if n is odd and r=n+1/2(2)
T∗r;n+1,1,0=0, if n is even, r=n/2+1 and n is replaced with n+1(3)
T∗r;n,1,0<>0, if n is even and r<>n/2(4)
T∗r;n,1,0<>0, if n is odd and r<>n+1/2Example 3.
LetY be a continuous r.v. that is nonnegative arising from EXPλ with CDF defined in (22); then, utilizing (58) and (59), we have(60)ζr;n,k,mY=−6+αT∗r;n,k,m4+αT∗r;n,k,m24λ,12ξY+Fr;n,k,m∗=17αT∗r;n,k,m6λ=EY24EY+Fr;n,k,m∗,12ξY+Rr;n,k,m∗=−3αT∗r;n,k,mλ=EY24EY+Rr;n,k,m∗.
For order statistics (withm=0 and k=1), we can apply some numerical values as follows:(1)
Forn=20, r=13, α=0.4, and λ=1, we have T∗r;n,1,0=5/21, ζr;n,1,0Y=−0.266251, and 1/2ξY+Fr;n,1,0∗=0.26984=EY2/4EY+Fr;n,1,0∗(2)
Forn=27, r=13, α=0.3, and λ=1, we have T∗r;n,1,0=−1/14, ζr;n,1,0Y=−0.246448, and1/2ξY+Rr;n,1,0∗=0.06428=EY2/4EY+Rr;n,1,0∗
which assure Theorem10.Example 4.
LetY be a continuous r.v. that is nonnegative arising from uniform distribution U0,b, b>0; then, utilizing (58) and (59), we have(61)ζr;n,k,mY=−b6010+5αT∗r;n,k,m+αT∗r;n,k,m2,12ξY+Fr;n,k,m∗=b8−1+10αT∗r;n,k,m,EY24EY+Fr;n,k,m∗=b12−1+15αT∗r;n,k,m,12ξY+Rr;n,k,m∗=−b243+32αT∗r;n,k,m,EY24EY+Rr;n,k,m∗=−b121+16αT∗r;n,k,m.
For order statistics (withm=0 and k=1), we can apply some numerical values as follows:(1)
Forn=20, r=13, α=0.4, and b=1, we have T∗r;n,1,0=5/21, ζr;n,1,0Y=−0.174754, 1/2ξY+Fr;n,1,0∗=−0.005952, and EY2/4EY+Fr;n,1,0∗=0.0357(2)
Forn=27, r=13, α=0.3, and b=1, we have T∗r;n,1,0=−1/14, ζr;n,1,0Y=−0.164889, and1/2ξY+Rr;n,1,0∗=−0.0964286,EY2/4EY+Rr;n,1,0∗=−0.0547619
Therefore, (1.) and (2.) assure Theorem10.Theorem 11.
LetY be a continuous r.v. that is nonnegative with survival function F¯Yy. From Remark 2, under the conditions on the parameters in (54), for order statistics (with m=0 and k=1) as a special case of m−gos, we denote(62)ζr;n,1,0Y<ζY+αT∗r;n,1,0s3+μ=Zr;n,1,0∗y,where si=mi+i∫mi∞F¯Yydy, mi=infy:F¯Yy≤1/i, i=1,2,…,n.Proof.
A method of obtaining an upper bound for the mean of the maximum ofn identically distributed nonnegative r.v.’s is given by Lai and Robbins [26] and Gravey [27]. If EYi+<∞, i=1,2,…,n, then(63)μn:n=EYn:n≤mn+n∫mn∞F¯Xxdx=sn.
Thus, from (54),(64)ζr;n,1,0Y<ζy+αT∗r;n,1,03EYFY2y+EY=ζy+αT∗r;n,1,0μ3:3+μ<ζY+αT∗r;n,1,0s3+μ=Zr;n,1,0∗y.Example 5.
SupposeY follows EXP1 with survival function F¯Yy=e−y,0<y. From (63), we have μn:n≤1+logn. Therefore, using (62) and from Remark 2, under the conditions on the parameters in (54), with n=50,r=28,α=0.3, and T∗r;n,1,0=5/51, we have ζr;n,1,0Y=−0.25493 and Zr;n,1,0∗y=−0.15886, which assure the previous results.


### 4.3. Stochastic Orders

Definition 3.
Y2 is known to be smaller than Y1 in the usual stochastic order, denoted by Y2≤stY1 if and only if F¯Y2t≤F¯Y1t, for all t∈ℝ. For more details, see Shaked and Shanthikumar [28].Theorem 12.
LetY1 and Y2 be two nonnegative continuous r.v.’s with CDF’s FY1⋅ and FY2⋅ and finite mean EY1 and EY2, respectively. If Y2≤stY1, then we have the following:(1)
From Remark2, under the conditions on the parameters in (54), we get(65)ζr;n,k,mY1−ζr;n,k,mY2≤αT∗r;n,k,mEY1−EY2.(2)
From Remark2, under the conditions on the parameters in (55), we get(66)ζr;n,k,mY1−ζr;n,k,mY2≤ζY1−ζY2.Proof.
First, from Theorem7 of Jahanshahi et al. [14], if Y2≤stY1, then ζY2≥ζY1. Therefore, the proof of (1) is(67)ζr;n,k,mY1−ζr;n,k,mY2≤ζY1−ζY2+3αT∗r;n,k,m∫0∞tFY12tfY1t−FY22tfY2tdt+αT∗r;n,k,mEY1−EY2≤αT∗r;n,k,mEY1−EY2.
The proof of (2) is(68)ζr;n,k,mY1−ζr;n,k,mY2≤ζY1−ζY2−4αT∗r;n,k,m∫0∞tFY1tfY1t−FY2tfY2tdt≤ζY1−ζY2.
Now, we will give an application of the last theorem as follows.Example 6.
LetY1 and Y2 be two r.v.’s of power function distribution with CDF’s FY1y=y2,0<y<1 and FY2y=y,0<y<1, respectively. For order statistics (with m=0 and k=1). From Theorem 12, we have the following:(1)
Forn=20, r=12, and α=0.6, we have T∗r;n,1,0=1/7, ζr;n,1,0Y1−ζr;n,1,0Y2=−0.09935, and αT∗r;n,k,mEY1−EY2=0.01428(2)
Forn=20, r=8, and α=0.8, we have T∗r;n,1,0=−5/21, ζr;n,1,0Y1−ζr;n,1,0Y2=−0.10121, and ζY1−ζY2=−0.1which assure the previous results.


## 4.1. Stop-Loss Transform and Gini Coefficient

In this section, we will presentζr;n,k,mY related to stop-loss transform and Gini index.Definition 1.
The stop-loss transformZFt of the nonnegative r.v. Y is defined as(47)ZFt=EmaxY−t,0=∫t∞F¯Yydy.Definition 2.
SupposeX and Y are independent r.v.’s and have the same distribution as X. Then, the Gini index or Gini coefficient is given by(48)Gindex=EX−YEX+Y=1−∫0∞F¯X2xdxEX,(see Wang [24] for more details).Remark 1.
From Jahanshahi et al. [14], based on (4) and Definitions 1 and 2, we have(49)ζY=−12EY−EZFY=−12EY−EF¯YymFY=EY2Gindex−1,where mFY=ZFY/F¯Yy is the mean residual life function.Theorem 8.
LetY be a continuous r.v. that is nonnegative with CREX ζX defined in (4). Then, the cumulative residual extropy of concomitants of m−gos, Yr;n,k,m, can be expressed as(50)ζr;n,k,mY=12EZFY+Tr;n,k,m∗=12EF¯YymFY+Tr;n,k,m∗=12EYGindex+Tr;n,k,m∗,where(51)Tr;n,k,m∗=αT∗r;n,k,m2−4αT∗r;n,k,mEYFYy+3αT∗r;n,k,m−αT∗r;n,k,m2EYFY2y+2αT∗r;n,k,m2EYFY3y+αT∗r;n,k,mEY.Proof.
The proof directly follows from (45) and Remark 1.


## 4.2. Upper Bound ofζr;n,k,mY

In another view of the cumulative residual extropy of concomitants ofm−gos, ζr;n,k,mY, we present it depending on its upper bound.Theorem 9.
LetY be a continuous r.v. that is nonnegative with survival function F¯Yy. Based on FGM family with CDF given by (9), utilizing (4), the residual extropy of the rth concomitants m−gos, Yr;n,k,m, 1≤r≤n, can be expressed by(52)ζr;n,k,mY≤ζy+αT∗r;n,k,m3EYFY2y+EY−4EYFYy.Proof.
Using Bernoulli’s inequality, from (44), we have(53)ζr;n,k,mY=−12∫0∞G¯r;n,k,my2dy=−12∫0∞F¯Y2y1+αT∗r;n,k,mFYy2dy≤−12∫0∞F¯Y2y1+2αT∗r;n,k,mFYydy=ζy+αT∗r;n,k,m3EYFY2y+EY−4EYFYy,where T∗r;n,k,m is defined in (12).
From the previous theorem, we can conclude the following remark.Remark 2.
Since−1<T∗r;n,k,m<1, then the residual extropy of the concomitant of rthm−gos given in (53) can be considered in following cases:(1)
IfT∗r;n,k,m=0, then ζr;n,k,mY=ζy.(2)
If0<α<1 and 0<T∗r;n,k,m<1 or −1<α<0 and −1<T∗r;n,k,m<0, then(54)ζr;n,k,mY<ζy+αT∗r;n,k,m3EYFY2y+EY.(3)
If0<α<1 and −1<T∗r;n,k,m<0 or −1<α<0 and 0<T∗r;n,k,m<1, then(55)ζr;n,k,mY<ζy−4αT∗r;n,k,mEYFYy.Theorem 10.
LetY be a continuous r.v. that is nonnegative with survival function F¯Yy. Then, we have the following:(1)
From Remark2, under the conditions on the parameters in (54), we get(56)ζr;n,k,mY<12ξY+Fr;n,k,m∗≤EY24EY+Fr;n,k,m∗.(2)
From Remark2, under the conditions on the parameters in (55), we get(57)ζr;n,k,mY<12ξY+Rr;n,k,m∗≤EY24EY+Rr;n,k,m∗,where ξY=−∫0∞F¯YylogF¯Yydy is the cumulative residual entropy presented by Rao et al. [25]:(58)Fr;n,k,m∗=3αT∗r;n,k,mEYFY2y+EYαT∗r;n,k,m−12,(59)Rr;n,k,m∗=−4αT∗r;n,k,mEYFYy−12EY.Proof.
The proof directly follows from inequality−logx≥1−x,x>0 and ξY≤EY2/2EY; see Rao et al. [25].
In the next remark, we will give an application of the previous results, taking the order statistics (withk=1 and m=0) as a special case under the FGM family.Remark 3.
For order statistic (m=0 and k=1), we have T∗r;n,1,0=−n−2r+1/n+1. Moreover, by controlling the values of n and r, we can observe the following:(1)
T∗r;n,1,0=0, if n is odd and r=n+1/2(2)
T∗r;n+1,1,0=0, if n is even, r=n/2+1 and n is replaced with n+1(3)
T∗r;n,1,0<>0, if n is even and r<>n/2(4)
T∗r;n,1,0<>0, if n is odd and r<>n+1/2Example 3.
LetY be a continuous r.v. that is nonnegative arising from EXPλ with CDF defined in (22); then, utilizing (58) and (59), we have(60)ζr;n,k,mY=−6+αT∗r;n,k,m4+αT∗r;n,k,m24λ,12ξY+Fr;n,k,m∗=17αT∗r;n,k,m6λ=EY24EY+Fr;n,k,m∗,12ξY+Rr;n,k,m∗=−3αT∗r;n,k,mλ=EY24EY+Rr;n,k,m∗.
For order statistics (withm=0 and k=1), we can apply some numerical values as follows:(1)
Forn=20, r=13, α=0.4, and λ=1, we have T∗r;n,1,0=5/21, ζr;n,1,0Y=−0.266251, and 1/2ξY+Fr;n,1,0∗=0.26984=EY2/4EY+Fr;n,1,0∗(2)
Forn=27, r=13, α=0.3, and λ=1, we have T∗r;n,1,0=−1/14, ζr;n,1,0Y=−0.246448, and1/2ξY+Rr;n,1,0∗=0.06428=EY2/4EY+Rr;n,1,0∗
which assure Theorem10.Example 4.
LetY be a continuous r.v. that is nonnegative arising from uniform distribution U0,b, b>0; then, utilizing (58) and (59), we have(61)ζr;n,k,mY=−b6010+5αT∗r;n,k,m+αT∗r;n,k,m2,12ξY+Fr;n,k,m∗=b8−1+10αT∗r;n,k,m,EY24EY+Fr;n,k,m∗=b12−1+15αT∗r;n,k,m,12ξY+Rr;n,k,m∗=−b243+32αT∗r;n,k,m,EY24EY+Rr;n,k,m∗=−b121+16αT∗r;n,k,m.
For order statistics (withm=0 and k=1), we can apply some numerical values as follows:(1)
Forn=20, r=13, α=0.4, and b=1, we have T∗r;n,1,0=5/21, ζr;n,1,0Y=−0.174754, 1/2ξY+Fr;n,1,0∗=−0.005952, and EY2/4EY+Fr;n,1,0∗=0.0357(2)
Forn=27, r=13, α=0.3, and b=1, we have T∗r;n,1,0=−1/14, ζr;n,1,0Y=−0.164889, and1/2ξY+Rr;n,1,0∗=−0.0964286,EY2/4EY+Rr;n,1,0∗=−0.0547619
Therefore, (1.) and (2.) assure Theorem10.Theorem 11.
LetY be a continuous r.v. that is nonnegative with survival function F¯Yy. From Remark 2, under the conditions on the parameters in (54), for order statistics (with m=0 and k=1) as a special case of m−gos, we denote(62)ζr;n,1,0Y<ζY+αT∗r;n,1,0s3+μ=Zr;n,1,0∗y,where si=mi+i∫mi∞F¯Yydy, mi=infy:F¯Yy≤1/i, i=1,2,…,n.Proof.
A method of obtaining an upper bound for the mean of the maximum ofn identically distributed nonnegative r.v.’s is given by Lai and Robbins [26] and Gravey [27]. If EYi+<∞, i=1,2,…,n, then(63)μn:n=EYn:n≤mn+n∫mn∞F¯Xxdx=sn.
Thus, from (54),(64)ζr;n,1,0Y<ζy+αT∗r;n,1,03EYFY2y+EY=ζy+αT∗r;n,1,0μ3:3+μ<ζY+αT∗r;n,1,0s3+μ=Zr;n,1,0∗y.Example 5.
SupposeY follows EXP1 with survival function F¯Yy=e−y,0<y. From (63), we have μn:n≤1+logn. Therefore, using (62) and from Remark 2, under the conditions on the parameters in (54), with n=50,r=28,α=0.3, and T∗r;n,1,0=5/51, we have ζr;n,1,0Y=−0.25493 and Zr;n,1,0∗y=−0.15886, which assure the previous results.


## 4.3. Stochastic Orders

Definition 3.
Y2 is known to be smaller than Y1 in the usual stochastic order, denoted by Y2≤stY1 if and only if F¯Y2t≤F¯Y1t, for all t∈ℝ. For more details, see Shaked and Shanthikumar [28].Theorem 12.
LetY1 and Y2 be two nonnegative continuous r.v.’s with CDF’s FY1⋅ and FY2⋅ and finite mean EY1 and EY2, respectively. If Y2≤stY1, then we have the following:(1)
From Remark2, under the conditions on the parameters in (54), we get(65)ζr;n,k,mY1−ζr;n,k,mY2≤αT∗r;n,k,mEY1−EY2.(2)
From Remark2, under the conditions on the parameters in (55), we get(66)ζr;n,k,mY1−ζr;n,k,mY2≤ζY1−ζY2.Proof.
First, from Theorem7 of Jahanshahi et al. [14], if Y2≤stY1, then ζY2≥ζY1. Therefore, the proof of (1) is(67)ζr;n,k,mY1−ζr;n,k,mY2≤ζY1−ζY2+3αT∗r;n,k,m∫0∞tFY12tfY1t−FY22tfY2tdt+αT∗r;n,k,mEY1−EY2≤αT∗r;n,k,mEY1−EY2.
The proof of (2) is(68)ζr;n,k,mY1−ζr;n,k,mY2≤ζY1−ζY2−4αT∗r;n,k,m∫0∞tFY1tfY1t−FY2tfY2tdt≤ζY1−ζY2.
Now, we will give an application of the last theorem as follows.Example 6.
LetY1 and Y2 be two r.v.’s of power function distribution with CDF’s FY1y=y2,0<y<1 and FY2y=y,0<y<1, respectively. For order statistics (with m=0 and k=1). From Theorem 12, we have the following:(1)
Forn=20, r=12, and α=0.6, we have T∗r;n,1,0=1/7, ζr;n,1,0Y1−ζr;n,1,0Y2=−0.09935, and αT∗r;n,k,mEY1−EY2=0.01428(2)
Forn=20, r=8, and α=0.8, we have T∗r;n,1,0=−5/21, ζr;n,1,0Y1−ζr;n,1,0Y2=−0.10121, and ζY1−ζY2=−0.1which assure the previous results.


## 5. Nonparametric Estimation

In this section, we obtain a nonparametric estimation of the residual extropy of concomitants ofm−gos under the FGM family by the empirical data. Let Y1,…,Yn be a random sample from a population with CDF F and its empirical estimator Fn. From (45), the empirical residual extropy of concomitants of m−gos is given by(69)ζr;n,k,mFn=−12∫0∞F¯n2y1+αT∗r;n,k,mFny2dy=−12∑j=1n−1∫YjYj+1F¯n2y1+αT∗r;n,k,mFny2dy,where F¯ny=1−Fny, Fny is the empirical CDF, and Y1≤Y2≤⋯≤Yn are the associated order statistics of the random sample.To estimateζr;n,k,m, we consider the first empirical estimator ζ1r;n,k,mFn as follows:(70)ζ1r;n,k,mFn=−12∑j=1n−1Wj+11−jn21+αT∗r;n,k,mjn2,where Wj+1=Yj+1−Yj, Fny=j/n, j=1,2,…,n−1, and W1=Y1. Moreover, the second empirical estimator (kernel-smoothed estimator) ζ2r;n,k,mFn is given by(71)ζ2r;n,k,mFn=−12∑j=1n−1Wj+11−Fnyj21+αT∗r;n,k,mFnyj2,where(72)Fnyj=1n∑i=1nhy−Yil,hy=∫−∞yKtdt and l is a bandwidth parameter; see Nadaraya [29].In the following examples, we apply the proposed methods to explain the performance of the empirical and kernel estimators.Example 7.
LetX1,…,Xn be a random sample of uniform distribution U0,1. According to Pyke [30], the sample spacing Wj+1 follows the beta distribution Beta1,n. Hence, from (70) and (71), we have(73)Eζ1r;n,k,mFn=−12n+1∑j=1n−11−jn21+αT∗r;n,k,mjn2,Varζ1r;n,k,mFn=n4n+12n+2∑j=1n−11−jn41+αT∗r;n,k,mjn4,Eζ2r;n,k,mFn=−12n+1∑j=1n−11−Fnyj21+αT∗r;n,k,mFnyj2,Varζ2r;n,k,mFn=n4n+12n+2∑j=1n−11−Fnyj41+αT∗r;n,k,mFnyj4.Example 8.
LetX1,…,Xn be a random sample of Expλ. According to Pyke [30], the sample spacing Wj+1 follows Expλn−j. Hence, from (70) and (71), we have(74)Eζ1r;n,k,mFn=−12nλ∑j=1n−11−jn1+αT∗r;n,k,mjn2,Varζ1r;n,k,mFn=14n2λ2∑j=1n−11−jn21+αT∗r;n,k,mjn4,Eζ2r;n,k,mFn=−12λ∑j=1n−11−Fnyj2n−j1+αT∗r;n,k,mFnyj2,Varζ2r;n,k,mFn=14λ2∑j=1n−11−Fnyj4n−j21+αT∗r;n,k,mFnyj4.Based on order statisticsk=1,m=0, Table 1 presents the mean and variance of ζ1r;n,1,0 and ζ2r;n,1,0 from U0,1 and EXP1, 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.Table 1
Mean and variance of the empirical residual extropy of concomitants of order statistics,α=0.8.
nrU0,1 distributionEXP1 distributionThe first estimatorThe second estimatorThe first estimatorThe second estimatorEζ1r;n,1,0Varζ2r;n,1,0Eζ2r;n,1,0Varζ2r;n,1,0Eζ1r;n,1,0Varζ2r;n,1,0Eζ2r;n,1,0Varζ2r;n,1,0103−0.10420.0019−0.10430.0013−0.17040.0047−0.20090.00575−0.12410.0024−0.13030.0019−0.21320.0065−0.25280.00848−0.15880.0035−0.175070.0035−0.29040.0108−0.34630.01453010−0.13170.00109−0.133010.000998−0.19770.0019−0.20890.001915−0.15120.001319−0.15440.00125−0.23730.0025−0.25520.002725−0.19660.00193−0.20390.00197−0.33330.00447−0.36630.00515015−0.13340.000702−0.134060.000667−0.19460.00116−0.200080.00110625−0.15720.00086−0.159060.00085−0.24230.0015−0.25390.001640−0.199060.001219−0.20260.00123−0.32940.0026−0.35140.00287020−0.13410.000515−0.13430.00049−0.19330.00082−0.19640.0007835−0.15980.00064−0.16090.00063−0.24450.00115−0.25260.0011660−0.21110.00096−0.21370.00097−0.35110.00208−0.36850.002210020−0.12650.00034−0.12590.00032−0.17670.00051−0.17760.0004850−0.16180.00046−0.16230.00045−0.24610.000814−0.25160.00081990−0.22060.00073−0.22260.00073−0.368090.00158−0.38070.0016


### 5.1. Data Application

In the following, we illustrate our empirical estimators in real and simulated data forEXP1.Example 9.
We refer to Kasilingam et al.’s [31] research to understand the spreading patterns of the COVID-19 virus; they used exponential growth modelling and identifies countries that have shown early signs of containment until 26th of March 2020. The data represent the percentage of serious cases of infections in 42 countries listed as follows: 1.56, 8.51, 2.17, 0.37, 1.09, 9.84, 4.95, 3.18, 11.37, 2.81, 6.22, 1.87, 0.00, 0.00, 9.05, 2.44, 1.38, 4.17, 3.74, 1.37, 2.33, 7.80, 2.10, 0.47, 2.54, 0.92, 0.09, 0.18, 1.72, 1.02, 0.62, 2.34, 0.50, 2.37, 3.65, 0.59, 5.76, 2.14, 0.88, 0.95, 4.17, and 2.25.We use Kolmogorov–Smirnov (K–S) test to check the fitting of the data for EXP1, which implies that the K-S statistic is 0.076282 with p value 0.9674. Thus, it is admitted to fit the data by EXP1; furthermore, see Figure 1. Based on EXP1, Figures 2 and 3 present the real-life and simulated data, respectively. Therefore, we can conclude that by decreasing α and increasing r, the empirical estimators approach the theoretical value and vice versa.Figure 1
Histogram of serious cases of infected COVID-19 virus data.Figure 2
Empirical estimators of serious cases of infected COVID-19 virus data. (a)α=0.8,n=40,r=35. (b) α=−0.8,n=40,r=35. (c) α=0.8,n=40,r=10. (d) α=−0.8,n=40,r=10.
(a)(b)(c)(d)Figure 3
Empirical estimators of simulated data. (a)α=0.8,n=100,r=90. (b) α=−0.8,n=100,r=90. (c) α=0.8,n=100,r=10. (d) α=−0.8,n=100,r=10.
(a)(b)(c)(d)


## 5.1. Data Application

In the following, we illustrate our empirical estimators in real and simulated data forEXP1.Example 9.
We refer to Kasilingam et al.’s [31] research to understand the spreading patterns of the COVID-19 virus; they used exponential growth modelling and identifies countries that have shown early signs of containment until 26th of March 2020. The data represent the percentage of serious cases of infections in 42 countries listed as follows: 1.56, 8.51, 2.17, 0.37, 1.09, 9.84, 4.95, 3.18, 11.37, 2.81, 6.22, 1.87, 0.00, 0.00, 9.05, 2.44, 1.38, 4.17, 3.74, 1.37, 2.33, 7.80, 2.10, 0.47, 2.54, 0.92, 0.09, 0.18, 1.72, 1.02, 0.62, 2.34, 0.50, 2.37, 3.65, 0.59, 5.76, 2.14, 0.88, 0.95, 4.17, and 2.25.We use Kolmogorov–Smirnov (K–S) test to check the fitting of the data for EXP1, which implies that the K-S statistic is 0.076282 with p value 0.9674. Thus, it is admitted to fit the data by EXP1; furthermore, see Figure 1. Based on EXP1, Figures 2 and 3 present the real-life and simulated data, respectively. Therefore, we can conclude that by decreasing α and increasing r, the empirical estimators approach the theoretical value and vice versa.Figure 1
Histogram of serious cases of infected COVID-19 virus data.Figure 2
Empirical estimators of serious cases of infected COVID-19 virus data. (a)α=0.8,n=40,r=35. (b) α=−0.8,n=40,r=35. (c) α=0.8,n=40,r=10. (d) α=−0.8,n=40,r=10.
(a)(b)(c)(d)Figure 3
Empirical estimators of simulated data. (a)α=0.8,n=100,r=90. (b) α=−0.8,n=100,r=90. (c) α=0.8,n=100,r=10. (d) α=−0.8,n=100,r=10.
(a)(b)(c)(d)


## 6. Conclusion

In this communication, we introduced the extropy ofm−gos and m−dgos arising from any distribution and wrote them in terms of the extropy of m−gos and m−dgos of U0,1 distribution, respectively. Furthermore, examples of the obtained models for exponential and Pareto distributions are provided. Also, we produced the lower bound of the extropy of m−gos emerging from any continuous distribution in terms of the extropy of m−gos of U0,1 distribution and the mode of the parent PDF. Moreover, residual and past extropy of m−gos and m−dgos are derived. Meanwhile, the residual extropy of concomitants of m−gos, ζr;n,k,mY, of FGM distribution is presented. The measure ζr;n,k,mY is discussed in terms of stop-loss transform and Gini coefficient. An alternative view of ζr;n,k,mY depending on its upper bound is considered. Besides, some examples and numerical results and stochastic orders of ζr;n,k,mY are obtained. Finally, we considered the problem of estimating ζr;n,k,mY 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.



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