EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7184 ISSN 1307-5543 – ejpam.com Published by New York Business Global Concomitant Extropy of Lai and Xie’s Extensions under Generalized Order Statistics: Properties, Estimation, and Application to Saudi Arabia Industrial Data Mohamed Said Mohamed1, S. M. EL-Arishy2, A. Gamal3, Saqer Abdullah Faqih4, R. A. Aldallal4,∗ 1 Department of Mathematics, College of Science and Humanities, Prince Sattam bin Abdulaziz University, Hawtat Bani Tamim 16511, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Al-Azhar University (Girls Branch), Cairo, Egypt 3 Department of Mathematics, Faculty of Science, Fayoum University, Egypt 4 Department of Management, College of Business Administration in Hawtat Bani Tamim, Prince Sattam bin Abdulaziz University, Saudi Arabia Abstract. In this work, we introduce and study the notion of concomitant extropy within the framework of generalized order statistics, extending the earlier contributions of Lai and Xie. The construction is supported with illustrative examples drawn from widely used probability distribu- tions, highlighting the flexibility and applicability of the proposed measure. Several recurrence relations and important special cases are derived, providing further insights into the structure of the model. In addition, we explore the behavior of past and residual extropies associated with the model, and extend the analysis to include negative cumulative extropy as well as cumulative residual extropy. To complement the theoretical findings, a non-parametric estimation procedure is developed and applied to real-world Saudi Arabia Industrial data, demonstrating the practical utility and relevance of the proposed approach. The results indicate that concomitant extropy can serve as a valuable tool for modeling and analyzing uncertainty in diverse applied settings. 2020 Mathematics Subject Classifications: 94A17, 62E99, 62P30 Key Words and Phrases: Concomitants, extropy, generalized order statistics, non-parametric estimation, Saudi Arabia industrial data ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7184 Email address: r.eldallal@psau.edu.sa (R. A. Aldallal) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. S. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7184 2 of 21 1. Introduction In industrial data analysis, statistical entropy is essential for quantifying the degree of uncertainty, randomness, or disorder present within complex datasets generated by mod- ern industrial systems. As industries increasingly rely on sensors, automation, and real time monitoring, the resulting data streams are often vast, multidimensional, and noisy. Traditional statistical measures such as variance or mean cannot fully capture the hidden patterns or unpredictability in such data. Statistical entropy provides a powerful metric to assess information content, detect anomalies, and evaluate system efficiency or stability. For instance, in manufacturing processes, higher entropy may indicate process instabil- ity or equipment malfunction, while lower entropy could reflect consistent and controlled operations. Thus, incorporating entropy based measures enables industries to enhance predictive maintenance, optimize resource allocation, and improve overall decision making in data driven industrial environments. Shannon [1] was the first to introduce information-theoretic entropy, which has since been applied in a wide range of fields, including computer science, medical research, and fi- nancial analysis. Lad et al. [2] showed that ”extropy” serves as a complementary dual functional to entropy. For a non-negative continuous random variable (r.v.) with proba- bility density function (PDF) g(y), the extropy is defined as: ε(Y ) = −1 2 ∫ ∞ 0 g2(y)dy. (1.1) Extropy was explored for order statistics (OS), record values, and some features by Qiu [3]. Through the use of extropy, Qiu et al. [4] introduced a mixed systems lifetime and derived its properties and limit of it. For additional research on extropy, see Yang et al. [5], Noughabi and Jarrahiferiz [6], Raqab and Qiu [7], Lad et al. [8], and Qiu and Jia [9]. Qiu and Jia [10] also looked into the meaning of residual extropy for a non-negative r.v. as εt(Y ) = εR(Y ; t) = −1 2Ḡ2(t) ∫ ∞ t g2(y)dy, t ≥ 0, (1.2) where Ḡ(t) = 1−G(t), G(t) is the cumulative distribution function (CDF). For the past lifetime of r.v. Yt = [t−Y |Y ≤ t], Krishnan et al. [11] provided the past extropy as follows εt(Y ) = εP (Y ; t) = −1 2G2(t) ∫ t 0 g2(y)dy. (1.3) From any continuous distribution, Jose and Sathar [[12], [13]] used the k-records’ residual and past extropies. Cumulative residual extropy (CREX) was proposed by Jahanshahi et M. S. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7184 3 of 21 al. [14]. The CREX is provided by a non-negative r.v. Y possessing a survival function Ḡ as follows ξ(Y ) = −1 2 ∫ ∞ 0 Ḡ2(y)dy. (1.4) Jahanshahi et al. [14] showed that ζ(Y ) is always negative. Tahmasebi and Toomaj [15] recently proposed negative cumulative extropy (NCEX), which is comparable to (1.1) and is described as ξ∗(Y ) = 1 2 ∫ ∞ 0 (1−G2(y))dy. (1.5) The Farlie-Gumbel-Morgenstern family (FGM) is described by a parameter δ, and the marginal distribution functionsGX(x) andGY (y), it was primarily derived by Morgenstern [16]. By adding further parameters, Lai and Xie [17] suggested the CDF as G(x, y) = GX(x)GY (y) + δḠX(x)αḠY (y) αGX(x)λGY (y) λ, α, λ ≥ 1, (1.6) for 0≤ δ ≤ 1. The related PDF is g(x, y) = gX(x)gY (y)(1 + δḠX(x)α−1ḠY (y) α−1GX(x)λ−1GY (y) λ−1 [λ− (α+ λ)GX(x)][λ− (α+ λ)GY (y)]). (1.7) According to Bairamov and Kotz [18], a bivariate copula for δ satisfies a broader range of conditions where: min{ 1 [K+(α, λ)]2 , 1 [K−(α, λ)]2 } ≤ δ ≤ 1 K+(α, λ)K−(α, λ) , with noting that K+ and K− are functions of α and λ. Furthermore, the conditional CDF and PDF are: GY |X(y | x) = GY (y) + δḠX(x)αḠY (y) αGX(x)λ−1GY (y) λ, α, λ ≥ 1, (1.8) gY |X(y | x) = gY (y)(1 + δḠX(x)α−1ḠY (y) α−1GX(x)λ−1GY (y) λ−1 × [λ− (α+ λ)GX(x)][λ− (α+ λ)GY (y)]). (1.9) Kamps [19] introduced the concept of generalized order statistics, with the special case of w-generalized order statistics (w-GOS), which provides a flexible framework for obtaining other ordered variables by choosing specific values of w. Gamal et al. [20] derived the M. S. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7184 4 of 21 CDF and PDF of the concomitant Y[r,n,w,l] from the Lai and Xie extensions for the r-th w-GOS, respectively, as: G[r,n,w,l](y) = GY (y) [ 1 + δη∗[r,n,w,l]ḠY (y) αGλ−1 Y (y) ] , (1.10) g[r,n,w,l](y) = gY (y) [ 1 + δR∗ [r,n,w,l]ḠY (y) α−1Gλ−1 Y (y) (λ− (α+ λ)GY (y)) ] , (1.11) and Ḡ[r,i,w,l](y) = 1−GY (y) [ 1 + δη∗[r,i,w,l]ḠY (y) αGλ−1 Y (y) ] = ḠY (y) [ 1 + δη∗[r,i,w,l]ḠY (y) α−1Gλ Y (y) ] . (1.12) where η∗[r,n,w,l] = mr−1 λ−1∑ ϵ=0 ( λ− 1 ϵ ) (−1)ϵ∏r q=1(γq + α+ ϵ) , (1.13) and R∗ [r,n,w,l] = mr−1[ λ−1∑ ϵ=0 λ ( λ− 1 ϵ ) (−1)ϵ∏r q=1(γq + ϵ+ α− 1) − λ∑ j=0 (α+ λ) ( λ j ) (−1)j∏r q=1(γq + j + α− 1) ], (1.14) with parameters l ≥ 1, n ∈ N, w1 = w2 = . . . = wn−1 = w, and 1 ≤ r ≤ n − 1, where γq = l + (n− q)(w + 1) and mr−1 = ∏r q=1 γq. For more details on the concept of w-GOS, see Kamps [19]. Extropy is regarded as the dual counterpart of entropy in both information theory and thermodynamics. Whereas entropy quantifies disorder, randomness, or uncertainty within a system, extropy emphasizes order, structure, and information content. It is often linked to concepts such as progress, growth, and the natural tendency of systems to evolve toward higher levels of organization. Because of these properties, extropy has recently attracted considerable attention as a complementary measure to entropy in modeling uncertainty and information. Within the FGM family, Almasoor et al. [21] investigated extropy measures for w-GOS concomitants, establishing beneficial properties and highlighting their flexibility. More recently, Mohamed et al. [[22], [23]] employed both real and simulated COVID-19 viral data to explore the non-parametric computation of the residual extropy measure for w- GOS concomitants, demonstrating the potential of extropy-based approaches in practical applications. M. S. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7184 5 of 21 Entropy for w-GOS concomitants arising from the Lai and Xie extensions was studied by Gamal et al. [20], who provided several theoretical insights. In contrast, the present work focuses on the study of extropy in this setting, extending the analysis to include several new measures and estimation methods. The main contributions and structure of this paper can be summarized as follows. In Section 2, we derive the expression for ε(Y[r,n,w,l]) using the extropy measure. In the same section, we also present conclusions concerning the extropy of the concomitants of order statistics and record values for the uniform and exponential distributions. In addition, the Lai and Xie extensions are employed to study several related measures, including residual and past extropies, CREX, and NCEX for w-GOS concomitants. Section 3 introduces empirical estimators for CREX, supported by simulation studies that validate the accuracy and efficiency of the proposed estimators. Real-life data are further analyzed to illustrate the practical applicability of non-parametric CREX estimation under the Lai and Xie framework. Finally, Section 4 provides a summary of the key findings and discusses possible directions for future research. 2. Lai and Xie extensions via extropy The measures of extropy, residual and past extropies, CREX, and NCEX for w-GOS concomitants will be derived in this section using the general framework of the Lai and Xie extensions. Furthermore, applications will be provided for the concomitants of OS and record values under the uniform and exponential distributions. 2.1. Extropy of concomitants for w-generalized order statistics In this subsection, we will derive the measure of extropy of concomitants for w-GOS, Y[r,n,w,l], from Lai and Xie extensions through the application of the theorem. Theorem 2.1.1. Assume (X,Y ) is a bivariate r.v. from the Lai and Xie extensions that is non-negative. Based on the concomitant Y[r,n,w,l], the rth w-GOS’s extropy is thus provided by ε(Y[r,n,w,l]) = ε(Y )− δR∗ [r,n,w,l]E((1− u)α−1uλ−1(λ− (α+ λ)u)g(G−1 Y (u))) − (δR∗ [r,n,w,l]) 2 2 E((1− u)2(α−1)u2(λ−1)(λ− (α+ λ)u)2g(G−1 Y (u))), (2.1) where ε(Y ) is extropy for r.v. Y and U is an uniformly distributed on U(0, 1). Proof. From (1.1) and (1.11), we have M. S. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7184 6 of 21 ε(Y[r,n,w,l]) = −1 2 ∫ ∞ 0 g2(r,n,w,l)(y)dy = −1 2 ∫ ∞ 0 g2Y (y) [ 1 + δR∗ [r,n,w,l]ḠY (y) α−1Gλ−1 Y (y) (λ− (α+ λ)GY (y)) ]2 dy = ε(Y )− δR∗ [r,n,w,l] ∫ ∞ 0 g2Y (y)ḠY (y) α−1Gλ−1 Y (y) (λ− (α+ λ)GY (y)) dy − (δR∗ [r,n,w,l]) 2 2 ∫ ∞ 0 g2Y (y)ḠY (y) 2(α−1)G 2(λ−1) Y (y) (λ− (α+ λ)GY (y)) 2 dy = ε(Y )− δR∗ [r,n,w,l] ∫ 1 0 (1− u)α−1 uλ−1 (λ− (α+ λ)u) g(G−1 Y (u))du − (δR∗ [r,n,w,l]) 2 2 ∫ 1 0 (1− u)2(α−1) u2(λ−1) (λ− (α+ λ)u)2 g(G−1 Y (u))du = ε(Y )− δR∗ [r,n,w,l]E((1− u)α−1 uλ−1 (λ− (α+ λ)u) g(G−1 Y (u))) − (δR∗ [r,n,w,l]) 2 2 E((1− u)2(α−1) u2(λ−1) (λ− (α+ λ)u)2 g(G−1 Y (u))), Theorem 2.1.2. Suppose that w = 0 and l = 1, the w-Gos reduces to OS. Therefore, the PDF of concomitants of OS from Lai and Xie extensions is given by g[r:n](y) = gY (y) [ 1 + δT ∗ [r:n]ḠY (y) α−1Gλ−1 Y (y) (λ− (α+ λ)GY (y)) ] , (2.2) where T ∗ [r:n] = n! (r − 1)!(n− r)! β(α+ n− r, λ+ r − 1) λ(n− r) + α(1− r) α+ n+ λ− 1 . (2.3) Proof: Since the PDF of the r-th OS is given by g[r:n](x) = n! (r − 1)!(n− r)! [1−GX(x)]n−r Gr−1 X (x)gX(x). (2.4) From (1.9) and (2.4), thus the OS’s concomitant PDF is provided by g[r:n](y) = ∫ ∞ −∞ gY |X(y | x)g[r:n](x)dx = gY (y) + δ n! (r − 1)!(n− r)! ḠY (y) α−1Gλ−1 Y (y) (λ− (α+ λ)GY (y)) gY (y) × ∫ ∞ −∞ [ λGX(x)λ+r−2 − (λ+ α)GX(x)λ+r−1 ] GX(x)n−r+α−1gX(x)dx. Let t = GX(x), then we have g[r:n](y) = gY (y) + δ n! (r − 1)!(n− r)! ḠY (y) α−1Gλ−1 Y (y) (λ− (α+ λ)GY (y)) gY (y) × ∫ 1 0 [ λtn−r+α−1(1− t)λ+r−2 − (λ+ α)tn−r+α−1(1− t)λ+r−1 ] dt. M. S. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7184 7 of 21 which proves the theorem. Corollary 2.1.1. According to Theorem (2.1.1), under the rth concomitants OS’s extropy, ε(Y[r:n]), is given by ε(Y[r:n]) = ε(Y )− δT ∗ [r:n]E((1− u)α−1uλ−1(λ− (α+ λ)u)g(G−1 Y (u))) − (δT ∗ [r:n]) 2 2 E((1− u)2(α−1)u2(λ−1)(λ− (α+ λ)u)2g(G−1 Y (u))). (2.5) Therefore, we have ε(Y[n:n]) = ε(Y )− δ αn(1− n) α+ λ+ n− 1 β(α, λ+ n− 1)E((1− u)α−1uλ−1(λ− (α+ λ)u)g(G−1 Y (u)))) − 1 2 { δαn(1− n) α+ λ+ n− 1 β(α, λ+ n− 1)}2E((1− u)2(α−1)u2(λ−1)(λ− (α+ λ)u)2g(G−1 Y (u)))), ε(Y[1:n]) = ε(Y )− δ λn(n− 1) α+ λ+ n− 1 β(α+ n− 1, λ)E((1− u)α−1uλ−1(λ− (α+ λ)u)g(G−1 Y (u))) − 1 2 { δλn(n− 1) α+ λ+ n− 1 β(α+ n− 1, λ)}2E((1− u)2(α−1)u2(λ−1)(λ− (α+ λ)u)2g(G−1 Y (u))). Remark 2.1. We get a recurrence relation between extropy for concomitants of OS from Lai and Xie extensions Y[r+1:n] and Y[r:n], then we have the result from corollary (2.1.1) as ε(Y[r+1:n])− ε(Y[r:n]) = δΩr,nE((1− u)α−1uλ−1(λ− (α+ λ)u)g(G−1 Y (u))) − δ2 (Ωr,n) 2 − 2T ∗ [r:n]Ωr,n 2 E((1− u)2(α−1)u2(λ−1)(λ− (α+ λ)u)2g(G−1 Y (u))), where Ωr,n = T ∗ [r:n] − T ∗ [r+1:n] = c(r−1:n){ λ(n− r) + α(1− r) (n− r)(α+ n+ λ− 1) β(α+ n− r, λ+ r − 1) − (λ(n− r − 1)− αr) r(α+ n− λ− 1) β(α+ n− r − 1, λ+ r)}, and c(r−1:n) = n! (r − 1)!(n− r − 1)! . Remark 2.2. We get a recurrence relation between extropy for concomitants of OS from Lai and Xie extensions Y[r:n−1] and Y[r:n], then we have the result from corollary (2.1.1) M. S. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7184 8 of 21 as ε(Y[r:n−1])− ε(Y[r:n]) = δΩ∗ r,nE((1− u)α−1uλ−1(λ− (α+ λ)u)g(G−1 Y (u))) − δ2 (Ω∗ r,n) 2 − 2T ∗ [r:n−1]Ω ∗ r,n 2 E((1− u)2(α−1)u2(λ−1)(λ− (α+ λ)u)2g(G−1 Y (u))), (2.6) where Ω∗ r,n = T ∗ [r:n] − T ∗ [r:n−1] = (n− 1)! (r − 1)!(n− r − 1)! {n(λ(n− r) + α(r − 1))(α+ n+ r − 1) (n− r)(α+ n+ λ)(α+ n+ λ− 1) − λ(n− r − 1) + α(r − 1) α+ n+ λ− 2 }β(α+ n− r − 1, λ+ r − 1). Theorem 2.1.3. The record value is a special case of the w-GOS with w = −1 and l = 1. Therefore, the PDF of concomitants of record value from Lai and Xie extensions is given by gL(n)(y) = gY (y) [ 1 + δµ∗ L(n)ḠY (y) α−1Gλ−1 Y (y) (λ− (α+ λ)GY (y)) ] , (2.7) where µ∗ L(n) = α−1∑ ϵ=0 ( α− 1 ϵ ) (−1)ϵ[ α+ λ (−(λ+ ϵ) + 1)n − λ (−(λ+ ϵ))n ]. (2.8) Proof: Since the PDF of the nth lower record is given by gL(n)(x) = 1 (n− 1)! [− lnGX(x)]n−1 gX(x). (2.9) From (1.9) and (2.9), then the PDF of the concomitant of the nth lower record is given by gL(n)(y) = ∫ ∞ −∞ gY |X(y | x)gL(n)(x)dx = gY (y) + δ (n− 1)! ḠY (y) α−1Gλ−1 Y (y) (λ− (α+ λ)GY (y)) gY (y) × ∫ ∞ −∞ ḠX(x)α−1Gλ−1 X (x) (λ− (α+ λ)GX(x)) [− lnGX(x)]n−1 gX(x)dx, M. S. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7184 9 of 21 let t = [− lnGX(x)], then we have gL(n)(y) = gY (y) + δ (n− 1)! ḠY (y) α−1Gλ−1 Y (y) (λ− (α+ λ)GY (y)) gY (y) × ∫ 1 0 [ (λ+ α)e(−λ+1)t − λe−λt ] (1− e−t)α−1tn−1dt. = gY (y) + δ (n− 1)! ḠY (y) α−1Gλ−1 Y (y) (λ− (α+ λ)GY (y)) gY (y) × α−1∑ ϵ=0 ( α− 1 ϵ ) (−1)ϵ ∫ 1 0 [ (λ+ α)e(−(λ+ϵ)+1)t − λe−(λ+ϵ)t ] tn−1dt. which proves the theorem. Corollary 2.1.2. According to Theorem (2.1.1), the concomitant extropy of YL(n) is given by ε(YL(n)) = ε(Y )− δµ∗ L(n)E((1− u)α−1uλ−1(λ− (α+ λ)u)g(G−1 Y (u))) − 1 2 (δµ∗ L(n)) 2E((1− u)2(α−1)u2(λ−1)(λ− (α+ λ)u)2g(G−1 Y (u))). (2.10) Now, we will give an application of Theorem (2.1.1) by the following examples. Example 2.1.1. Assume that the exponential distribution (Exp(θ)) with CDF produces the non-negative continuous r.v. Y as follows G(y) = 1− e−θy, θ > 0, y > 0. (2.11) According to Theorem (2.1.1), the concomitant extropy ε(Y[r,n,w,l]) is given by ε(Y[r,n,w,l]) = −0.25θ − 3δR∗ [r,n,w,l]θΓ(3 + α)Γ(1 + λ) Γ(4 + α+ λ) − (δR∗ [r,n,w,l]) 2θ × λ(10λ+ α(4 + α+ λ))Γ(4 + 2α)Γ(−1 + 2λ) Γ(5 + 2α+ 2λ) . (2.12) Based on OS, Figure 1 shows some plots of ε(Y[r,n,0,1]) for Exp(θ). Example 2.1.2. Assume that the uniform distribution with CDF produces the non-negative continuous r.v. Y as follows G(y) = ( y σ ) , 0 < y < σ. (2.13) M. S. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7184 10 of 21 Figure 1: plots of ε(Y[r,n,0,1]) form Exp(θ) for a sample of size n = 200, a = 1, b = 2 with various selections: (a) r = 30, θ1 = 1, θ2 = 2, θ3 = 3, (b) θ = 2, r1 = 30, r2 = 150, r3 = 170. According to Theorem (2.1.1), the concomitant extropy ε(Y[r,n,w,l]) is given by ε(Y[r,n,w,l]) = −0.5 σ − 2δR∗ [r,n,w,l]Γ(2 + α)Γ(1 + λ) σΓ(3 + α+ λ) − (δR∗ [r,n,w,l]) 2 σ × λ(6λ+ α(3 + α+ λ))Γ(3 + 2α)Γ(−1 + 2λ) Γ(2(2 + α+ λ)) . (2.14) Based on OS, Figure 2 shows some plots of ε(Y[r,n,0,1]) for a uniform distribution Figure 2: plots of ε(Y[r,n,0,1]) form a uniform distribution for a sample of size n = 200, α = 1, λ = 2 with various selections: (c) r = 30, σ1 = 0.5, σ2 = 1, σ3 = 2, σ4 = 3, (d) σ = 2, r1 = 30, r2 = 80, r3 = 100, r4 = 110. M. S. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7184 11 of 21 2.2. Residual and past extropies of concomitants for w-generalized order statistics In this subsection, we will derive the measures of the residual and past extropies of w-GOS, Y[r,n,w,l], from Lai and Xie extensions by following theorems. Theorem 2.2.1. Assume that the r-th w-GOS concomitant Y[r,n,w,l] is a continuous and non-negative r.v. from the Lai and Xie extensions. Then, the residual extropy of the r-th w-GOS, based on the concomitant Y[r,n,w,l], is given by εt(Y[r,n,w,l]) =εR(Y[r,n,w,l]; t) = [ 1 1 + δη∗[r,n,w,l]ḠY (t)α−1Gλ Y (t) ]2[εR(Y ) − δR∗ [r,n,w,l] Ḡ2 Y (t) E((1− u)α−1 uλ−1g(G−1 Y (u)) (λ− (α+ λ)u)) − (δR∗ [r,n,w,l]) 2 2Ḡ2 Y (t) E((1− u)2(α−1) u2(λ−1) (λ− (α+ λ)u)2 g(G−1 Y (u)))], where εR(Y ) is residual extropy for r.v. of Y and UR is r.v. from U(1, GY (t)). Proof. From (1.2), (1.11) and (1.12), we have εR(Y[r,n,w,l]; t) = −1 2 ∫ ∞ t g2Y (y) [ 1 + δR∗ [r,n,w,l]ḠY (y) α−1Gλ−1 Y (y) (λ− (α+ λ)GY (y)) ]2 dy Ḡ2 Y (t) [ 1 + δη∗[r,n,w,l]ḠY (t)α−1Gλ Y (t) ]2 = [ 1 1 + δη∗[r,n,w,l]ḠY (t)α−1Gλ Y (t) ]2[εR(Y )− δR∗ [r,n,w,l] Ḡ2 Y (t) ∫ ∞ t g2Y (y) × ḠY (y) α−1Gλ−1 Y (y) (λ− (α+ λ)GY (y)) dy − (δR∗ [r,n,w,l]) 2 2Ḡ2 Y (t) ∫ ∞ t g2Y (y) × ḠY (y) 2(α−1)G 2(λ−1) Y (y) (λ− (α+ λ)GY (y)) 2 dy]. = [ 1 1 + δη∗[r,n,w,l]ḠY (t)α−1Gλ Y (t) ]2[εR(Y )− δR∗ [r,n,w,l] Ḡ2 Y (t) × ∫ 1 GY (t) (1− u)α−1 uλ−1 (λ− (α+ λ)u) g(G−1 Y (u))du− (δR∗ [r,n,w,l]) 2 2Ḡ2 Y (t) × ∫ 1 GY (t) (1− u)2(α−1) u2(λ−1) (λ− (α+ λ)u)2 g(G−1 Y (u))du] M. S. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7184 12 of 21 = [ 1 1 + δη∗[r,n,w,l]ḠY (t)α−1Gλ Y (t) ]2[εR(Y )− δR∗ [r,n,w,l] Ḡ2 Y (t) × E((1− u)α−1 uλ−1 (λ− (α+ λ)u) g(G−1 Y (u)))− (δR∗ [r,n,w,l]) 2 2Ḡ2 Y (t) × E((1− u)2(α−1) u2(λ−1) (λ− (α+ λ)u)2 g(G−1 Y (u)))]. Theorem 2.2.2. Assume that the r-th w-GOS concomitant Y[r,n,w,l] is a continuous and non-negative r.v. from the Lai and Xie extensions. From (1.3), (1.10), and (1.11), the residual extropy of the r-th w-GOS, based on the concomitant Y[r,n,w,l], is given by εt(Y[r,n,w,l]) = εP (Y[r,n,w,l]; t) = [ 1 1 + δη∗[r,n,w,l]ḠY (t)αG λ−1 Y (t) ]2[εP (Y ) − δR∗ [r,n,w,l] G2 Y (t) E((1− u)α−1 uλ−1 (λ− (α+ λ)u) g(G−1 Y (u))) − (δR∗ [r,n,w,l]) 2 2G2 Y (t) E((1− u)2(α−1) u2(λ−1) (λ− (α+ λ)u)2 g(G−1 Y (u)))], where εP (Y ) is past extropy for r.v. of Y and UP is r.v. from U(0, GY (t)). Proof. An analogous set of steps to those used in the proof of Theorem 2.2.1 could be applied here. 2.3. Cumulative residual extropy of concomitants for w-generalized order statistics In this subsection, we will derive the measure of CREX of concomitants for w-GOS, Y[r,n,w,l], from Lai and Xie extensions by following theorem. Theorem 2.3.1. Assume that the r-th w-GOS concomitant Y[r,n,w,l] is a continuous and non-negative r.v. from the Lai and Xie extensions. Then, the CREX extropy of the r-th w-GOS, based on the concomitant Y[r,n,w,l], is given by ξ[r,n,w,l](Y ) = ξ(Y )− δη∗[r,n,w,l]E((1− u)α+1uλ 1 g(G−1 Y (u)) )− (δη∗[r,n,w,l]) 2 2 × E((1− u)2αu2λ 1 g(G−1 Y (u)) ). M. S. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7184 13 of 21 Proof. From (1.4) and (1.12), we have ξ[r,n,w,l](Y ) = −1 2 ∫ ∞ 0 (Ḡ[r,n,w,l](y)) 2dy = −1 2 ∫ ∞ 0 [ Ḡ2 Y (y) + 2δη∗[r,n,w,l]ḠY (y) α+1Gλ Y (y) + (δη∗[r,n,w,l]) 2ḠY (y) 2αG2λ Y (y) ] dy. = ξ(Y )− δη∗[r,n,w,l] ∫ ∞ 0 (1− u)α+1uλ 1 g(G−1 Y (u)) du− (δη∗[r,n,w,l]) 2 2 × ∫ ∞ 0 (1− u)2αu2λ 1 g(G−1 Y (u)) du, with noting that ξ(Y ) is the CREX of r.v. Y, and U is r.v. from U(0, 1). 2.4. The negative cumulative extropy of concomitants for w-generalized order statistics In this subsection, we will derive the measure of NCEX of concomitants for w-GOS, Y[r,n,w,l], from Lai and Xie extensions by following theorem. Theorem 2.4.1. Assume that the r-th w-GOS concomitant Y[r,n,w,l] is a continuous and non-negative r.v. from the Lai and Xie extensions. Then, the NCEX extropy of the r-th w-GOS, based on the concomitant Y[r,n,w,l], is given by ξ∗(Y ) = ξ∗(Y ) + δη∗[r,n,w,l]E((1− u)αuλ+1 1 g(G−1 Y (u)) ) + (δη∗[r,n,w,l]) 2 2 × E((1− u)2αu2λ 1 g(G−1 Y (u)) ). Proof. From (1.5) and (1.10), we have ξ∗(Y ) = 1 2 ∫ ∞ 0 (1−G2 [r,n,w,l](y))dy = 1 2 ∫ ∞ 0 [ (1−G2 Y (y)) + 2δη∗[r,n,w,l]ḠY (y) αGλ+1 Y (y) + (δη∗[r,n,w,l]) 2ḠY (y) 2αG2λ Y (y) ] dy = ξ∗(Y ) + δη∗[r,n,w,l] ∫ ∞ 0 (1− u)αuλ+1 1 g(G−1 Y (u)) du+ (δη∗[r,n,w,l]) 2 2 × ∫ ∞ 0 (1− u)2αu2λ 1 g(G−1 Y (u)) du, with noting that ξ∗(Y ) is the NCEX of r.v. Y, and U is r.v. on U(0, 1). M. S. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7184 14 of 21 3. Non-parametric estimation Using the empirical data, we derive a non-parametric estimate of the CREX of w- GOS concomitants under the Lai and Xie extensions in this section. Consider the random sample Y1, ..., Yn from a population with CDF and its empirical estimator Gn. According to (1.4) and (1.12), the empirical CREX of w-GOS concomitants is provided by ξ[r,n,w,l](Gn) = −1 2 ∫ ∞ 0 Ḡ2 n(y)(1 + δη∗[r,n,w,l]G λ n(y)(1−Gn(y)) α−1))2dy = −1 2 n−1∑ j=1 ∫ Y(j+1) Y(j) Ḡ2 n(y)(1 + δη∗[r,n,w,l]G λ n(y)(1−Gn(y)) α−1))2dy, (3.1) where Ḡn(y) = 1−Gn(y), and the corresponding OS of the random sample is Y(1) ≤ Y(2) ≤ ... ≤ Y(n), whereas Gn(y) is the empirical CDF. . We take into consideration the first empirical estimator ξ1[r,n,w,l](Gn) in order to esti- mate ξ[r,n,w,l] as follows ξ1[r,n,w,l](Gn) = −1 2 n−1∑ j=1 Wj+1 ( 1− j n )2( 1 + δη∗[r,n,w,l]( j n )λ(1− j n )α−1 )2 , (3.2) where Gn(y) = j n , j = 1, 2, ..., n − 1, Wj+1 = Y(j+1) − Y(j), W1 = Y(1). Additionally, the kernel-smoothed estimator, or second empirical estimator, ξ2[r,n,w,l](Gn), is provided by ξ2[r,n,w,l](Gn) = −1 2 n−1∑ j=1 Wj+1 (1−Gn(yj)) 2 ( 1 + δη∗[r,n,w,l](Gn(yj)) λ(1−Gn(yj)) α−1 )2 , (3.3) where Gn(yj) = 1 n n∑ i=1 B ( y − Yi h ) , B(y) = ∫ y −∞ S(t)dt and h are bandwidth parameters; refer to Nadaraya [24]. By using 100 samples of size n=25, are generated with the uniform distribution U(0, 1). These data satisfy the asymptomatic normality of the empirical estimator’s assumption in (3.2) and (3.3). The histogram shown in Figure 3 is presented. M. S. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7184 15 of 21 (a) First Estimator Sample Values D en si ty −3 −2 −1 0 1 2 0. 0 0. 1 0. 2 0. 3 0. 4 Standard Normal Distribution (b) second Estimator Sample Values D en si ty −2 −1 0 1 2 3 0. 0 0. 1 0. 2 0. 3 0. 4 Standard Normal Distribution Figure 3: Histogram of empirical estimators for sample values under Lai and Xie extensions at α = 3, λ = 4, δ = 0.6 with standard normal distribution. In the following, we use the suggested techniques in the examples to describe how the empirical and kernel estimators work. Example 3.0.1. Consider the random sample X1, ..., Xn with U(0, 1). Pyke states in [25] that the sample spacing Wj+1 is in accordance with the beta distribution Beta(1, n). Thus, derived from (3.2) and (3.3), we obtain E(ξ1[r,n,w,l](Gn)) = −1 2(1 + n) n−1∑ j=1 ( 1− j n )2( 1 + δη∗[r,n,w,l]( j n )λ(1− j n )α−1 )2 , V ar(ξ1[r,n,w,l](Gn)) = n 4(2 + n)(1 + n)2 n−1∑ j=1 ( 1− j n )4( 1 + δη∗[r,n,w,l]( j n )λ(1− j n )α−1 )4 , E(ξ2[r,n,w,l](Gn)) = −1 2(1 + n) n−1∑ j=1 (1−Gn(yj)) 2 ( 1 + δη∗[r,n,w,l](Gn(yj)) λ(1−Gn(yj)) α−1 )2 , V ar(ξ2[r,n,w,l](Gn)) = n 4(2 + n)(1 + n)2 n−1∑ j=1 (1−Gn(yj)) 4 ( 1 + δη∗[r,n,w,l](Gn(yj)) λ(1−Gn(yj)) α−1 )4 . Example 3.0.2. Consider the random sample X1, ..., Xn with Exp(θ). Pyke states in [25] that the sample spacing Wj+1 is in accordance with Exp(θ(n − j)). Thus, derived from (3.2) and (3.3), we get E(ξ1[r,n,w,l](Gn)) = −1 2nθ n−1∑ j=1 ( 1− j n )( 1 + δη∗[r,n,w,l]( j n )λ(1− j n )α−1 )2 , M. S. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7184 16 of 21 V ar(ξ1[r,n,w,l](Gn)) = 1 4n2θ2 n−1∑ j=1 ( 1− j n )2( 1 + δη∗[r,n,w,l]( j n )λ(1− j n )α−1 )4 , E(ξ2[r,n,w,l](Gn)) = −1 2θ n−1∑ j=1 (1−Gn(yj)) 2 n− j ( 1 + δη∗[r,n,w,l](Gn(yj)) λ(1−Gn(yj)) α−1 )2 , V ar(ξ2[r,n,w,l](Gn)) = 1 4θ2 n−1∑ j=1 (1−Gn(yj)) 4 (n− j)2 ( 1 + δη∗[r,n,w,l](Gn(yj)) λ(1−Gn(yj)) α−1 )4 . Based on OS, Table (1) presents the mean and variance of ξ1[r;n,0,1] and ξ2[r;n,0,1] from EXP (θ), For varying sample size values (n = 10, 30, 60, 100) using (θ = 0.5, 1, 2). Table 1: The empirical estimators’ mean and variance for CREX of concomitants of OS under Lai and Xie extensions at α = 5, λ = 7, r = 3, and δ = 0.8. n θ EXP (θ) distribution The first estimator The second estimator E(ξ1[r;n,0,1]) V ar(ξ1[r;n,0,1]) E(ξ2[r;n,0,1]) V ar(ξ2[r;n,0,1]) 10 0.5 -0.4500 0.0285 -0.5543 0.0387 1 -0.2250 0.0071 -0.2837 0.0095 2 -0.1125 0.0017 -0.1538 0.0027 30 0.5 -0.4833 0.0105 -0.5178 0.0115 1 -0.2416 0.0026 -0.2600 0.0028 2 -0.1208 0.0006 -0.1325 0.0006 60 0.5 -0.4916 0.0054 -0.5088 0.0056 1 -0.2458 0.0013 -0.2550 0.0013 2 -0.1229 0.0003 -0.1290 0.0004 100 0.5 -0.4950 0.0032 -0.5054 0.0033 1 -0.2475 0.0008 -0.2531 0.0008 2 -0.1237 0.0002 -0.1278 0.0001 Table 1 provides the characteristics listed below: (i) When n is a fixed value, the values of the mean increase as the values of θ increase, whereas the values of variance decrease with θ. M. S. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7184 17 of 21 (ii) For a fixed θ, both the values of mean and variance are declining as n increases in value. (iii) When n approaches infinity, the value of the variance tends to zero. Simulated data are shown in Figure 4. Thus, we can infer that the empirical estimators go closer to the theoretical value as r and θ increase, and vice versa. 0 20 40 60 80 − 1. 0 − 0. 8 − 0. 6 − 0. 4 − 0. 2 0. 0 (a) Sample Size E m pi ric al e st im at es 0 20 40 60 80 − 1. 0 − 0. 8 − 0. 6 − 0. 4 − 0. 2 0. 0 First estimator Second estimator Theoretical estimates 30 40 50 60 70 80 − 1. 0 − 0. 8 − 0. 6 − 0. 4 − 0. 2 0. 0 (b) Sample Size E m pi ric al e st im at es 30 40 50 60 70 80 − 1. 0 − 0. 8 − 0. 6 − 0. 4 − 0. 2 0. 0 First estimator Second estimator Theoretical estimates 0 20 40 60 80 − 1. 0 − 0. 8 − 0. 6 − 0. 4 − 0. 2 0. 0 (c) Sample Size E m pi ric al e st im at es 0 20 40 60 80 − 1. 0 − 0. 8 − 0. 6 − 0. 4 − 0. 2 0. 0 First estimator Second estimator Theoretical estimates 30 40 50 60 70 80 − 1. 0 − 0. 8 − 0. 6 − 0. 4 − 0. 2 0. 0 (d) Sample Size E m pi ric al e st im at es 30 40 50 60 70 80 − 1. 0 − 0. 8 − 0. 6 − 0. 4 − 0. 2 0. 0 First estimator Second estimator Theoretical estimates Figure 4: Empirical estimators of simulated data for CREX of concomitants of OS under Lai and Xie extensions. (a)α = 5, λ = 7, n = 80, r = 3, δ = 0.8, θ = 0.5. (b)α = 5, λ = 7, n = 80, r = 30, δ = 0.8, θ = 0.5. (c)α = 5, λ = 7, n = 80, r = 3, δ = 0.8, θ = 1. (d)α = 5, λ = 7, n = 80, r = 30, δ = 0.8, θ = 1. 3.1. Saudi Arabia Real industrial data analysis The Industrial Production Index (IPI) is a key economic indicator that measures the monthly output of factories, mines, and utilities within an economy. It reflects the real volume of industrial production and provides valuable insights into the performance of M. S. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7184 18 of 21 the industrial sector, which is a major driver of overall economic growth. A higher IPI value typically indicates an expansion in industrial activity, suggesting that factories are producing more goods to meet rising demand, which in turn contributes to employment and economic development. Conversely, a lower IPI value points to reduced industrial ac- tivity, signaling a potential slowdown in the economy due to weaker demand or production challenges. In Saudi Arabia, the IPI serves as an important measure for assessing the progress of the Kingdom’s industrial diversification efforts under Vision 2030, particularly in sectors such as manufacturing, mining, and energy. The data presented below, obtained from the General Authority for Statistics https://www.stats.gov.sa/en/statistics-tabs? tab=436312&category=123454 , shows the monthly index for the manufacture of basic metals from January 2023 to August 2025: 116.6, 106.3, 119.6, 111.3, 113.8, 120.4, 122.1, 120.8, 114.3, 121.3, 117.0, 117.1, 120.7, 123.7, 123.4, 131.4, 127.7, 125.9, 125.1, 124.8, 122.4, 123.0, 121.8, 118.8, 115.9, 111.1, 112.4, 113.6, 109.5, 107.7, 108.6, 108.7 The Weibull(β1, β2) distribution was able to properly fit this data set with parameter β1 = 20.6709 and β2 = 120.993, which a p-value equal to 0.971468. Furthermore, see Figure 5, which shows the estimated PDF and CDF. Figure 5: Estimated PDF and CDF of Weibull distribution for data Based on the above real data, the empirical estimators for CREX of concomitants of OS under Lai and Xie extensions at α = 4, λ = 6, and δ = 0.8 will be ξ1[20;32,0,1] = −4.06006 and ξ2[20;32,0,1] = −4.26736. Meanwhile, the theoretical value is ξ[20;32,0,1] = −56.9977. As we can see from the previous data, the second estimator is better in representation to the theoretical than the first one is. https://www.stats.gov.sa/en/statistics-tabs?tab=436312&category=123454 https://www.stats.gov.sa/en/statistics-tabs?tab=436312&category=123454 M. S. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7184 19 of 21 4. Conclusions In this work, we have investigated several measures of extropy, including residual and past extropy, CREX, and NCEX, in the context of the concomitant of w-GOS based on the extensions of Lai and Xie. A number of new properties of extropy related to record values and order statistics were derived and discussed. In addition, empirical estimation techniques were employed to estimate CREX, and the performance of the proposed esti- mators was assessed through both numerical illustrations for the exponential distribution EXP (θ) and simulation studies. The simulation results demonstrate that the empirical estimators converge toward the theoretical values as both r and θ increase. Moreover, the application to real-life data, originally considered in the Lai and Xie framework, provided further evidence of the practical relevance of the proposed methodology. In particular, the non-parametric estimation of CREX for w-GOS concomitants showed that the kernel-smoothed estimator, corresponding to the second empirical estimator, consistently outperformed alternatives across all values of n and r. Furthermore, the results revealed that the mean squared error decreases as the sample size n and the order r increase, confirming the efficiency of the estimators. Overall, the findings of this study enrich the theoretical understanding of extropy- related measures within generalized order statistics and highlight their potential appli- cability in practical data analysis. Future work may focus on extending these concepts to broader distributional families, exploring their connections with other information- theoretic measures, and developing further applications in reliability analysis, survival studies, and statistical modeling. Acknowledgements The authors extend their appreciation to Prince Sattam bin Abdulaziz University for funding this research work through the project number (PSAU/2025/02/32732) Author Contributions Each author contributed equally to the study’s inception and design. Competing Interests The writers are not required to disclose any relevant financial or non-financial interests. M. S. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7184 20 of 21 Ethics approval The authors approve of all the ethical issues. Data Availability The article contains the datasets created and/or analyzed during the present investi- gation. Declarative use of AI techniques The authors confirm that AI tools were not utilized in crafting this article. Conflict of interest The authors confirm the lack of any conflict of interest. References [1] C. E. Shannon. A mathematical theory of communication. The Bell System Technical Journal, 27(3):379–423, 1948. [2] F. Lad, G. Sanfilippo, and G. Agro. Extropy: Complementary dual of entropy. Statistical Science, pages 40–58, 2015. [3] G. Qiu. The extropy of order statistics and record values. Statistics and Probability Letters, 120:52–60, 2017. [4] G. Qiu, L. Wang, and X. Wang. On extropy properties of mixed systems. Probability in the Engineering and Informational Sciences, 33(3):471–486, 2019. [5] J. Yang, W. Xia, and T. Hu. Bounds on extropy with variational distance constraint. Probability in the Engineering and Informational Sciences, 33(2):186–204, 2019. [6] H. Alizadeh Noughabi and J. Jarrahiferiz. On the estimation of extropy. Journal of Non-parametric Statistics, 31(1):88–99, 2019. [7] M. Z. Raqab and G. Qiu. On extropy properties of ranked set sampling. Statistics, 53(1):210–226, 2019. [8] F. Lad, G. Sanfilippo, and G. Agrò. The duality of entropy/extropy, and completion of the kullback information complex. Entropy, 20(8):593, 2018. [9] G. Qiu and K. Jia. Extropy estimators with applications in testing uniformity. Journal of Non-parametric Statistics, 30(1):182–196, 2018. [10] G. Qiu and K. Jia. The residual extropy of order statistics. Statistics and Probability Letters, 133:15–22, 2018. M. S. Mohamed et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7184 21 of 21 [11] A. S. Krishnan, S. M. Sunoj, and N. Unnikrishnan Nair. Some reliability properties of extropy for residual and past lifetime random variables. Journal of the Korean Statistical Society, 49:457–474, 2020. [12] J. Jose and E. A. Sathar. Residual extropy of k-record values. Statistics and Proba- bility Letters, 146:1–6, 2019. [13] E. A. Sathar and J. Jose. Past extropy of k-records. Stochastics and Quality Control, 35(1):25–38, 2020. [14] S. M. A. Jahanshahi, H. Zarei, and A. H. Khammar. On cumulative residual extropy. Probability in the Engineering and Informational Sciences, 34(4):605–625, 2020. [15] S. Tahmasebi and A. Toomaj. On negative cumulative extropy with applications. Communications in Statistics-Theory and Methods, 51(15):5025–5047, 2022. [16] D. Morgentern. Einfache beispiele zweidimensionaler verteilunngen. Mitteilungsblatt für Mathematische Statistik, 8:234–235, 1956. [17] C. D. Lai and M. Xie. A new family of positive quadrant dependent bivariate distri- butions. Statistics and Probability Letters, 46(4):359–364, 2000. [18] I. Bairamov, S. Kotz, and M. Bekci. New generalized farlie-gumbel-morgenstern distri- butions and concomitants of order statistics. Journal of Applied Statistics, 28(5):521– 536, 2001. [19] U. Kamps. A concept of generalized order statistics, volume 48 of Journal of Statistical Planning and Inference. Elsevier, 1995. [20] A. Gamal, L. S. Diab, and M. S. Mohamed. Concomitants for case-i of generalized order statistical and its dual from lai and xie extension. Pakistan Journal of Statistics, 40(1), 2024. [21] Z. Almaspoor, A. A. Jafari, and S. Tahmasebi. Measures of extropy for concomitants of generalized order statistics in morgenstern family. Journal of Statistical Theory and Applications, 21(1):1–20, 2022. [22] M. S. Mohamed, A. T. Abdulrahman, Z. Almaspoor, and M. Yusuf. Ordered variables and their concomitants under extropy via covid-19 data application. Complexity, 2021(1):6491817, 2021. [23] M. S. Mohamed, H. M. Barakat, S. A. Alyami, and M. A. Abd Elgawad. Cumulative residual tsallis entropy-based test of uniformity and some new findings. Mathematics, 10(5):771, 2022. [24] E. A. Nadaraya. On estimating regression. Theory of Probability and Its Applications, 9(1):141–142, 1964. [25] R. Pyke. Spacings. Journal of the Royal Statistical Society: Series B (Methodological), 27(3):395–436, 1965. Introduction Lai and Xie extensions via extropy Extropy of concomitants for w-generalized order statistics Residual and past extropies of concomitants for w-generalized order statistics Cumulative residual extropy of concomitants for w-generalized order statistics The negative cumulative extropy of concomitants for w-generalized order statistics Non-parametric estimation Saudi Arabia Real industrial data analysis Conclusions