# The Extropy of Concomitants of Generalized Order Statistics from Huang–Kotz–Morgenstern Bivariate Distribution

Authors: I. A. Husseiny; A. H. Syam
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we study the extropy for concomitants ofm−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 of m−GOS is used to investigate the problem of estimating the CREX and NCEX. The concomitants of order statistics and record values are offered as some applications of these findings.


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## Body

## 1. Introduction

The classic Farlie–Gumbel–Morgenstern (FGM) family of the marginalsFX and FY was treated by Huang and Kotz [1] as a polynomial-type single parameter extension. The distribution function (DF) which they suggested is(1)FX,Yx,y=FXxFYy1+λ1−FXpx1−FYpy,p>0.The corresponding probability density function (PDF) is given by(2)fX,Yx,y=fXxfYy1+λ1−1+pFXpx⋅1−1+pFYpy,denoted by HK-FGM. The allowable range of the association parameter λ is −1/p2≤λ≤1/p, and the range for correlation coefficient is −p+2−2min1,p2≤ρ≤3pp+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 fields, e.g., Abd Elgawad et al. [2], Bairamov and Kotz [3], Barakat et al. [4], and Fisher and Klein [5].Kamps [6] proposed a unified 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 γi=k+n−im+1,i=1,2,…,n. The RVs Xr,n,m,k,r=1,2,…,n, are said to be m−GOSs based on the DF FXx, if their joint PDF is of the form(3)f1,2,…,nx1,x2,…xn=k∏j=1n−1γj∏i=1n−1F¯mxifxiF¯k−1xnfxn,F−11≥xn≥…≥x1≥F−10. The marginal PDF of rth m−GOS, Xr,n,m,k, 1≤r≤n, is given by (cf. [6])(4)fXr,n,m,kx=Cr−1r−1!F¯Xγr−1xfXxgmr−1FXx,where F¯X=1−FX, Cr−1=∏i=1rγi, r=1,2,…,n, gmx=hmx−hm1,x∈0,1, and hmx=−1−1−xm+1/m+1, if m≠−1, while h−1x=−log1−x.David [7] proposed the concept of concomitants of OSs for the first time. The 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. [14–16], and Tahmasebi et al. [17] have studied the concomitants of GOSs. Assume that Xi,Yi,i=1,2,…,n, is a random sample from the given bivariate DF FX,Yx,y. If the X–variates are arranged in ascending order as X1,n,m,k≤X2,n,m,k≤…≤Xn,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 Yr,n,m,k,r=1,2,…,n. The PDF of the concomitant of rth m−GOS is given by (see, e.g., Abd Elgawad et al. [2] and Alawady et al. [14])(5)fr,n,m,ky=∫−∞∞fY|Xy|xfXr,n,m,kxdx,where fY|Xy|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(6)JX=−12∫0∞fXx2dx=−12∫01fXFX−1udu.It is obvious thatJX≤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. The 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. The CREX is defined as(7)ξ∗X=−12∫0∞F¯X2xdx.It is always negative. As a result, the negative CREX (NCREX) can be expressed as(8)ξX=12∫0∞F¯X2xdx.Newly, Tahmasebi and Toomaj [28] suggested a negative cumulative extropy (NCEX) analogous to (8), defined as(9)CξX=12∫0∞1−FX2xdx=∫01ϕufXFX−1udu,where ϕu=1−u2/2,0<u<1.In this paper, we aim to investigate the extropy for concomitants ofm−GOSs in the HK-FGM family. Therefore, the rest of this paper is organized as follows. In Section 2, we begin by calculating the extropy measure of Yr,n,m,k in the HK-FGM family. Some results of CREX and NCEX for Yr,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.


## 2. Extropy and Some of Its Related Measures in Concomitants ofm−GOSs Based on HK-FGM

In the present section, we derive the measures of extropy, CREX, and NCEX for the concomitantYr,n,m,k of m−GOSs based on HK-FGM family. Some of these measures properties are revealed.


### 2.1. Extropy in Concomitants ofm−GOSs from HK-FGM

LetX∼FX and Y∼FY. Since the conditional PDF of Yr,n,m,k given Xr,n,m,k=x is fYr,n,m,k|Xr,n,m,ky|x=fY|Xy|x (cf. Abd Elgawad et al. [2]), we can easily find the PDF of Yr,n,m,k in m− GOS as follows:(10)fr,n,m,ky=∫−∞∞fY|Xy|xfXr,n,m,kxdx=∫−∞∞fYy1+λ1−1+pFXpx1−1+pFYpyCr−1r−1!F¯Xγr−1xgmr−1FXx.dFX=fYy1+1−1+pFYpyλ1−1+pCr−1r−1!∫−∞∞FXpxF¯Xγr−1xgmr−1FXxdFXx=fYy1+1−1+pFYpyλ1−1+pCr−1r−1!∫−∞∞1−1−FXxpF¯Xγr−1xgmr−1FXxdFXx=fYy1+1−1+pFYpyλ1−1+pCr−1r−1!∑j=0φp−1j∫−∞∞F¯Xγr+j−1xgmr−1FXxdFXx,where φx=∞, if x is noninteger, and φx=x, if x is integer. Now, by making the transformation t=1−F¯Xm+1x/m+1 (see [29]), then we get(11)fr,n,m,ky=fYy+λCr,n,m,k:p∗fYy−fVy=fYy1+λCr,n,m,k:p∗1−1+pFYPy,where fVy is the PDF of the RV V∼FYp+1 and Cr,n,m,k:p∗=1−1+p∑j=0φp−1j∏i=1rγi/γi+j.Therefore, the DF and survival function ofYr,n,m,k follow(12)Fr,n,m,ky=FYy1+λCr,n,m,k:p∗1−FYPy,(13)F¯r,n,m,ky=F¯Yy−λCr,n,m,k:p∗FYy1−FYPy.IfYr,n,m,k is the concomitant of the m−GOS with PDF (11), then the extropy measure is given by(14)JYr,n,m,k=−12∫0∞fr,n,m,k2ydy=−12∫0∞fy1+λCr,n,m,k:p∗1−1+pFYpy2dy=−12∫0∞f2y1+2λCr,n,m,k:p∗1−1+pFYpy+λCr,n,m,k:p∗21−21+pFYpy+1+p2FY2pydy=1+λCr,n,m,k:p∗2JY+1+pλCr,n,m,k:p∗+λCr,n,m,k:p∗2EfYF−1UUp−1+p22λCr,n,m,k:p∗2EfYF−1UU2p,where U is a uniformly RV on 0,1, and JY is the extropy of the RV Y. Assume that Qu=FY−1u is the quantile function. The quantile density function is defined as qu=1/fYQu, where qu is the derivative of Qu with respect to u, i.e., qu=Q′u. Therefore, the corresponding quantile-based JYr,n,m,k can be written as(15)JYr,n,m,k=1+λCr,n,m,k:p∗2JY+1+pλCr,n,m,k:p∗+λCr,n,m,k:p∗2EUpqu−1+p22λCr,n,m,k:p∗2EU2pqu=JY−λCr,n,m,k:p∗22E1−1+pUp2qu−λCr,n,m,k:p∗E1−1+pUpqu.Remark 1.
Ifm=0 and k=1, the m-GOSs turn into OSs. The PDF and CDF of the concomitant of the rth OS Yr:n are given by(16)fr:ny=fYy1+λΔr,n:p1−1+pFYpy,and(17)Fr:ny=FYy1+λΔr,n:p1−FYpy,respectively, where Δr,n:p=1−1+pβr+p,n−r+1/βr,n−r+1. According to (15), the extropy measure of Yr:n is(18)JYr:n=JY−λΔr,n:p22E1−1+pUp2qu−λΔr,n:pE1−1+pUpqu.
Therefore,(19)JYn:n=JY−λp−np/n+p22E1−1+pUp2qu−λp−npn+pE1−1+pUpqu,and(20)JY1:n=JY−λ1−1+pp!n!/n+p!22E1−1+pUp2qu−λ1−1+pp!n!n+p!E1−1+pUpqu.Theorem 1.
LetYr:n be the concomitant of the rth OS from HK-FGM; then, we have(21)JYr+1:n−JYr:n=λΩr,n:pE1−1+pUpqu−λΩr,n:p2−2λ2Δr,n:pΩr,n:p2E1−1+pUp2qu,and(22)JYr+2:n−JYr:n=λ2r+p+1r+1Ωr,n:pE1−1+pUpqu−λ2r+p+1r+1Ωr,n:pE1−1+pUpqu,where Ωr,n:p=pp+1βr+p,n−r+1/rβr,n−r+1.Proof.
It is easy to check that(23)Δr+1,n:p=Δr,n:p−pp+1βr+p,n−r+1rβr,n−r+1,and(24)Δr+2,n:p=Δr,n:p−pp+12r+b+1βr+p,n−r+1rr+1βr,n−r+1.Theorem 2.
LetYr:n be the concomitant of the rth OS from HK-FGM; then, we have(25)JYr:n−1−JYr:n=λΩr,n:p∗E1−1+pUpqu−λΩr,n:p∗2−2λ2Δr,n:pΩr,n:p∗2E1−1+pUp2qu,and(26)JYr:n−2−JYr:n=λ2n+p−1n−1Ωr,n:p∗E1−1+pUpqu−λ2n+p−1/n−1Ωr,n:p∗2−2λ22n+p−1/n−1Δr,n:pΩr,n:p∗2E1−1+pUp2qu,where Ωr,n:p∗=pp+1βr+p,n−r+1/nβr,n−r+1.Proof.
It is easy to check that(27)Δr,n−1:p=Δr,n:p−pp+1βr+p,n−r+1nβr,n−r+1,and(28)Δr,n−2:p=Δr,n:p−pp+12n+b−1βr+p,n−r+1nn−1βr,n−r+1.Example 1.
Assume thatX and Y have a joint exponential distribution as HK-FGM (denoted by HK-FGM-ED)(29)FX,Yx,y=1−exp−xθ11−exp−yθ21+λ1−1−exp−xθ1p1−1−exp−yθ2p,x,y>0.
By using (15), we get the extropy in Yr:n as(30)JYr:n=−14θ2−λΔr,n:p223p2θ24p2+10p+4−λΔr,n:ppθ22p+4,θ2>0.Example 2.
Assume thatX and Y have a joint logistic distribution as HK-FGM(31)FX,Yx,y=1+exp−x−11+exp−y−11+λ1−1+exp−x−p1−1+exp−y−p,−∞<x<∞,−∞<y<∞.
By using (15), we have(32)JYr:n=−112−λΔr,n:p2216−21+pp2+5p+6+p+124p2+10p+6−λΔr,n:p16−1+pp2+5p+6.Example 3.
Assume thatX and Y have a joint exponentiated exponential distribution as HK-FGM(33)FX,Yx,y=1−exp−θ1xα11−exp−θ2yα21+λ1−1−exp−θ1xα1p1−1−exp−θ2yα2p.
Then, we have(34)JYr:n=θ2α241−2α2−λΔr,n:p22−θ2α221−2α2−21+pθ2α2p+2α2p+1−1+p+12θ2α22p+12α2p+1−1−λΔr,n:p−θ2α221−2α2−1+pθ2α2p+2α2p+1−1.Remark 2.
Assume thatXi,Yi,i=1,2,…, is a sequence of bivariate RVs drawn from a continuous DF of a random vector X,Y. The value of Y that corresponds to the rth upper record Rr,r>1, pertaining to the variate X will be referred to as the rth concomitant of the record value Rr and it will be denoted by Rr. The record values and their corresponding concomitants are applicable in real-world experiments like lifetime studies, sporting events, and other experimental fields. The 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]. The record value is a special case of the m−GOSs with m=−1 and k=1. Therefore, the PDF and DF for Rr are obtained as(35)gry=fYy1+λCr:p1−1+pFYpy,Gry=FYy1+λCr:p1−FYpy,where Cr:p=1−1+p∑i=0φp−1ipi/i+1r. Therefore, the extropy measure for Rr is obtained as follows:(36)JRr=JY−λCr:p22E1−1+pUp2qu−λCr:pE1−1+pUpqu.


### 2.2. CREX in Concomitants ofm−GOSs Based on HK-FGM

The CREX measures are provided for the concomitantYr,n,m,k of the rth m−GOS as follows:(37)ξ∗Yr,n,m,k=−12∫0∞F¯r,n,m,k2ydy=−12∫0∞F¯Yy−λCr,n,m,k:p∗FYy1−FYPy2dy=ξ∗Y+λCr,n,m,k:p∗∫0∞FYyF¯Yy1−FYpydy−λCr,n,m,k:p∗22∫0∞FY2y1−FYpy2dy=ξ∗Y+λCr,n,m,k:p∗EU1−U1−Upqu−λCr,n,m,k:p∗22EU21−Up2qu.Example 4.
For the HK-FGM bivariate uniform distribution (HK-FGM-UD) with DF(38)FX,Yx,y=xyθ1θ21+λ1−xθ1p1−yθ2p,0<x<θ1,0<y<θ2.
By using (37), we get the CREX in Yr:n as(39)ξ∗Yr:n=−θ26+λΔr,n:pθ2pp+56p+2p+3−λΔr,n:p2θ2p232p2+9p+9.


### 2.3. NCEX in Concomitants ofm−GOSs from HK-FGM

For the concomitant ofm−GOS, the NCEX is given by(40)CξYr,n,m,k=12∫0∞1−Fr,n,m,k2ydy=12∫0∞1−FYy1+λCr,n,m,k:p∗1−FYPy2dy=CξY−λCr,n,m,k:p∗EU21−Upqu−λCr,n,m,k:p∗22EU21−Up2qu,where CξY is the NCEX of the RV Y.


## 2.1. Extropy in Concomitants ofm−GOSs from HK-FGM

LetX∼FX and Y∼FY. Since the conditional PDF of Yr,n,m,k given Xr,n,m,k=x is fYr,n,m,k|Xr,n,m,ky|x=fY|Xy|x (cf. Abd Elgawad et al. [2]), we can easily find the PDF of Yr,n,m,k in m− GOS as follows:(10)fr,n,m,ky=∫−∞∞fY|Xy|xfXr,n,m,kxdx=∫−∞∞fYy1+λ1−1+pFXpx1−1+pFYpyCr−1r−1!F¯Xγr−1xgmr−1FXx.dFX=fYy1+1−1+pFYpyλ1−1+pCr−1r−1!∫−∞∞FXpxF¯Xγr−1xgmr−1FXxdFXx=fYy1+1−1+pFYpyλ1−1+pCr−1r−1!∫−∞∞1−1−FXxpF¯Xγr−1xgmr−1FXxdFXx=fYy1+1−1+pFYpyλ1−1+pCr−1r−1!∑j=0φp−1j∫−∞∞F¯Xγr+j−1xgmr−1FXxdFXx,where φx=∞, if x is noninteger, and φx=x, if x is integer. Now, by making the transformation t=1−F¯Xm+1x/m+1 (see [29]), then we get(11)fr,n,m,ky=fYy+λCr,n,m,k:p∗fYy−fVy=fYy1+λCr,n,m,k:p∗1−1+pFYPy,where fVy is the PDF of the RV V∼FYp+1 and Cr,n,m,k:p∗=1−1+p∑j=0φp−1j∏i=1rγi/γi+j.Therefore, the DF and survival function ofYr,n,m,k follow(12)Fr,n,m,ky=FYy1+λCr,n,m,k:p∗1−FYPy,(13)F¯r,n,m,ky=F¯Yy−λCr,n,m,k:p∗FYy1−FYPy.IfYr,n,m,k is the concomitant of the m−GOS with PDF (11), then the extropy measure is given by(14)JYr,n,m,k=−12∫0∞fr,n,m,k2ydy=−12∫0∞fy1+λCr,n,m,k:p∗1−1+pFYpy2dy=−12∫0∞f2y1+2λCr,n,m,k:p∗1−1+pFYpy+λCr,n,m,k:p∗21−21+pFYpy+1+p2FY2pydy=1+λCr,n,m,k:p∗2JY+1+pλCr,n,m,k:p∗+λCr,n,m,k:p∗2EfYF−1UUp−1+p22λCr,n,m,k:p∗2EfYF−1UU2p,where U is a uniformly RV on 0,1, and JY is the extropy of the RV Y. Assume that Qu=FY−1u is the quantile function. The quantile density function is defined as qu=1/fYQu, where qu is the derivative of Qu with respect to u, i.e., qu=Q′u. Therefore, the corresponding quantile-based JYr,n,m,k can be written as(15)JYr,n,m,k=1+λCr,n,m,k:p∗2JY+1+pλCr,n,m,k:p∗+λCr,n,m,k:p∗2EUpqu−1+p22λCr,n,m,k:p∗2EU2pqu=JY−λCr,n,m,k:p∗22E1−1+pUp2qu−λCr,n,m,k:p∗E1−1+pUpqu.Remark 1.
Ifm=0 and k=1, the m-GOSs turn into OSs. The PDF and CDF of the concomitant of the rth OS Yr:n are given by(16)fr:ny=fYy1+λΔr,n:p1−1+pFYpy,and(17)Fr:ny=FYy1+λΔr,n:p1−FYpy,respectively, where Δr,n:p=1−1+pβr+p,n−r+1/βr,n−r+1. According to (15), the extropy measure of Yr:n is(18)JYr:n=JY−λΔr,n:p22E1−1+pUp2qu−λΔr,n:pE1−1+pUpqu.
Therefore,(19)JYn:n=JY−λp−np/n+p22E1−1+pUp2qu−λp−npn+pE1−1+pUpqu,and(20)JY1:n=JY−λ1−1+pp!n!/n+p!22E1−1+pUp2qu−λ1−1+pp!n!n+p!E1−1+pUpqu.Theorem 1.
LetYr:n be the concomitant of the rth OS from HK-FGM; then, we have(21)JYr+1:n−JYr:n=λΩr,n:pE1−1+pUpqu−λΩr,n:p2−2λ2Δr,n:pΩr,n:p2E1−1+pUp2qu,and(22)JYr+2:n−JYr:n=λ2r+p+1r+1Ωr,n:pE1−1+pUpqu−λ2r+p+1r+1Ωr,n:pE1−1+pUpqu,where Ωr,n:p=pp+1βr+p,n−r+1/rβr,n−r+1.Proof.
It is easy to check that(23)Δr+1,n:p=Δr,n:p−pp+1βr+p,n−r+1rβr,n−r+1,and(24)Δr+2,n:p=Δr,n:p−pp+12r+b+1βr+p,n−r+1rr+1βr,n−r+1.Theorem 2.
LetYr:n be the concomitant of the rth OS from HK-FGM; then, we have(25)JYr:n−1−JYr:n=λΩr,n:p∗E1−1+pUpqu−λΩr,n:p∗2−2λ2Δr,n:pΩr,n:p∗2E1−1+pUp2qu,and(26)JYr:n−2−JYr:n=λ2n+p−1n−1Ωr,n:p∗E1−1+pUpqu−λ2n+p−1/n−1Ωr,n:p∗2−2λ22n+p−1/n−1Δr,n:pΩr,n:p∗2E1−1+pUp2qu,where Ωr,n:p∗=pp+1βr+p,n−r+1/nβr,n−r+1.Proof.
It is easy to check that(27)Δr,n−1:p=Δr,n:p−pp+1βr+p,n−r+1nβr,n−r+1,and(28)Δr,n−2:p=Δr,n:p−pp+12n+b−1βr+p,n−r+1nn−1βr,n−r+1.Example 1.
Assume thatX and Y have a joint exponential distribution as HK-FGM (denoted by HK-FGM-ED)(29)FX,Yx,y=1−exp−xθ11−exp−yθ21+λ1−1−exp−xθ1p1−1−exp−yθ2p,x,y>0.
By using (15), we get the extropy in Yr:n as(30)JYr:n=−14θ2−λΔr,n:p223p2θ24p2+10p+4−λΔr,n:ppθ22p+4,θ2>0.Example 2.
Assume thatX and Y have a joint logistic distribution as HK-FGM(31)FX,Yx,y=1+exp−x−11+exp−y−11+λ1−1+exp−x−p1−1+exp−y−p,−∞<x<∞,−∞<y<∞.
By using (15), we have(32)JYr:n=−112−λΔr,n:p2216−21+pp2+5p+6+p+124p2+10p+6−λΔr,n:p16−1+pp2+5p+6.Example 3.
Assume thatX and Y have a joint exponentiated exponential distribution as HK-FGM(33)FX,Yx,y=1−exp−θ1xα11−exp−θ2yα21+λ1−1−exp−θ1xα1p1−1−exp−θ2yα2p.
Then, we have(34)JYr:n=θ2α241−2α2−λΔr,n:p22−θ2α221−2α2−21+pθ2α2p+2α2p+1−1+p+12θ2α22p+12α2p+1−1−λΔr,n:p−θ2α221−2α2−1+pθ2α2p+2α2p+1−1.Remark 2.
Assume thatXi,Yi,i=1,2,…, is a sequence of bivariate RVs drawn from a continuous DF of a random vector X,Y. The value of Y that corresponds to the rth upper record Rr,r>1, pertaining to the variate X will be referred to as the rth concomitant of the record value Rr and it will be denoted by Rr. The record values and their corresponding concomitants are applicable in real-world experiments like lifetime studies, sporting events, and other experimental fields. The 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]. The record value is a special case of the m−GOSs with m=−1 and k=1. Therefore, the PDF and DF for Rr are obtained as(35)gry=fYy1+λCr:p1−1+pFYpy,Gry=FYy1+λCr:p1−FYpy,where Cr:p=1−1+p∑i=0φp−1ipi/i+1r. Therefore, the extropy measure for Rr is obtained as follows:(36)JRr=JY−λCr:p22E1−1+pUp2qu−λCr:pE1−1+pUpqu.


## 2.2. CREX in Concomitants ofm−GOSs Based on HK-FGM

The CREX measures are provided for the concomitantYr,n,m,k of the rth m−GOS as follows:(37)ξ∗Yr,n,m,k=−12∫0∞F¯r,n,m,k2ydy=−12∫0∞F¯Yy−λCr,n,m,k:p∗FYy1−FYPy2dy=ξ∗Y+λCr,n,m,k:p∗∫0∞FYyF¯Yy1−FYpydy−λCr,n,m,k:p∗22∫0∞FY2y1−FYpy2dy=ξ∗Y+λCr,n,m,k:p∗EU1−U1−Upqu−λCr,n,m,k:p∗22EU21−Up2qu.Example 4.
For the HK-FGM bivariate uniform distribution (HK-FGM-UD) with DF(38)FX,Yx,y=xyθ1θ21+λ1−xθ1p1−yθ2p,0<x<θ1,0<y<θ2.
By using (37), we get the CREX in Yr:n as(39)ξ∗Yr:n=−θ26+λΔr,n:pθ2pp+56p+2p+3−λΔr,n:p2θ2p232p2+9p+9.


## 2.3. NCEX in Concomitants ofm−GOSs from HK-FGM

For the concomitant ofm−GOS, the NCEX is given by(40)CξYr,n,m,k=12∫0∞1−Fr,n,m,k2ydy=12∫0∞1−FYy1+λCr,n,m,k:p∗1−FYPy2dy=CξY−λCr,n,m,k:p∗EU21−Upqu−λCr,n,m,k:p∗22EU21−Up2qu,where CξY is the NCEX of the RV Y.


## 3. Estimating ofNCREX and NCEX

In this section, we use empirical estimators to calculate the NCREX and NCEX for concomitantYr,n,m,k. We will now look at the problem of using the empirical NCREX to estimate the NCREX for concomitants. Let Xi,Yi, where i=1,2,…, be a HK-FGM sequence. Using the relation −ξ∗=ξ, the empirical NCREX of Yr,n,m,k may be calculated as follows:(41)ξ^Yr,n,m,k=12∫0∞1−F^r,n,m,ky2dy=12∫0∞1−F^Yy−λCr,n,m,k:p∗F^Yy1−F^Ypy2dy=12∑j=1n−1∫ZjZj+11−F^Yy2−2λCr,n,m,k:p∗F^Yy1−F^Yy1−F^Ypy+λCr,n,m,k:p∗2F^Y2y1−F^Ypy2dy=12∑j=1n−1Uj1−jn2−2λCr,n,m,k:p∗jn1−jn1−jnp+λCr,n,m,k:p∗2jn1−jnp2,where Uj=Zj+1−Zj, j=1,2…,n−1, are the sample spacings based on ordered random samples Yj.Example 5.
IfXi,Yi, i=1,2,…,n, is a random sample from HK-FGM-ED, the sample spacings Uj are independent and exponentially distributed, with a mean 1/θ2n−j (for more details, see [30]). According to Pyke [33], the empirical NCREX expectation and variance based on Yr:n are as follows:(42)Eξ^Yr:n=12θ2∑j=1n−11n−j1−jn2−2λΔr,n:pjn1−jn1−jnpλΔr,n:p2jn1−jnp2,(43)Varξ^Yr:n=14θ22∑j=1n−11n−j21−jn2−2λΔr,n:pjn1−jn1−jnp+λΔr,n:p2jn1−jnp22.Example 6.
The sample spacingsUj for HK-FGM-UD with θ1=θ2=1 are independent of the beta distribution with parameters 1 and n. According to Pyke [33], the expectation and variance of empirical NCREX based on Rr are as follows:(44)Eξ^Rr=1n+1∑j=1n−11−jn2−2λCr:pjn1−jn1−jnp+λCr:p2jn1−jnp2,(45)Varξ^Rr=nn+12n+2∑j=1n−11−jn2−2λCr:pjn1−jn1−jnp+λCr:p2jn1−jnp22.
Also, the empirical NCEX ofYr,n,m,k can be obtained as follows:(46)ξ^Yr,n,m,k=12∫0∞1−F^r,n,m,k2ydy=12∫0∞1−F^Yy1+λCr,n,m,k:p∗1−F^YPy2dy=12∑j=1n−1∫ZjZj+11−F^Y2y−2λCr,n,m,k:p∗F^Y2y1−F^Ypy−λCr,n,m,k:p∗2F^Y2y1−F^Ypy2dy=12∑j=1n−1Uj1−jn2−2λCr,n,m,k:p∗jn21−jnp−λCr,n,m,k:p∗2jn1−jnp2.
Table1 displays the values of Eξ^Yr:n and Var ξ^Yr:n based on HK-FGM-ED by computing (42) and (43). The following properties can be extracted from Table 1.(1)
Generally, with fixedn and θ2, the values of Eξ^Yr:n and Var ξ^Yr:n decrease as the value of λ increases.(2)
Generally, with fixedn and θ2, the values of Eξ^Yr:n and Var ξ^Yr:n decrease as the value of θ2 increases.
Figure1 shows the relation between NCREX and empirical NCREX of Rr from HK-FGM-UD at n=100. The following properties can be extracted from Figure 1.(1)
NCREX and empirical NCREX have similar values whenp=1 and −1≤λ≤1 and especially when λ=0 for all values of r.(2)
The value of NCREX is greater than the value of empirical NCREX forp>1 and 0<λ≤1.Table 1
Eξ^Yr:n and Var ξ^Yr:n from HK-FGM-ED at r=2.
nθ2λ=−0.25λ=−0.15λ=0.15λ=0.5Eξ^Y2:nforp=250.50.46890.44130.35900.2642510.23450.22060.17950.1321520.11720.11030.08970.0661100.50.55770.51440.38620.23941010.27880.25720.19310.11971020.13940.12860.09650.0598150.50.58380.53670.39730.23791510.29190.26840.198611891520.14590.13420.09930.0595200.50.59580.54720.40340.23922010.29790.27360.20170.11962020.14890.13680.10090.0598Varξ^Y2:nforp=250.50.05800.05380.04280.0327510.01450.01350.01070.0082520.00630.00340.00270.0020100.50.03660.03310.02450.01751010.00920.00830.00610.00441020.00230.00210.00150.0011150.50.58380.53670.39730.23791510.29190.26840.198611891520.14590.13420.09930.0595200.50.59580.54720.40340.23922010.29790.27360.20170.11962020.14890.13680.10090.0598Figure 1
Representation of NCREX and empirical NCREX based onRr from HK-FGM-UD. (a) p = 1, λ = −1. (b) p = 1, λ = 1. (c) p = 2, λ = 0.5. (d) p = 2, λ = −0.25. (e) p = 3, λ = 0.3. (f) p = 3, λ = −0.1. (g) p = 2, λ = 0 and (h) p = 3, λ = 0.
(a)(b)(c)(d)(e)(f)(g)(h)



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