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Research article

Spatio-temporal alterations, configurations, and distribution of green areas, along with their sustainability in Parakou, Benin

  • Green areas (GAs) are swiftly declining in urban areas worldwide, amplifying adverse local climate impacts on the well-being of city residents. Despite this, there is limited empirical research on the changing patterns and distribution of GAs and their vulnerability. This is especially notable in dry tropical cities where these spaces function as vital microclimate areas that control against climate change effects such as flooding and heat islands. This study focused on examining the changing GA coverage, scrutinizing the spatial distribution of different GA categories, and investigating threat factors associated with their perceived sustainability in Parakou. Employing a mixed-methods approach, open-source geospatial data and collected primary data were acquired through on-site observations as well as semi-structured interviews. Data analysis involved the application of geospatial, statistical, and textual techniques. The results indicated that, from 2000 to 2020, the city experienced a loss of 16.48 km2 (24.73%) in its GA cover. The predominant land use change observed was the conversion of sparse vegetation (21.86%) into built-up areas. A notable difference (P < 0.0001) was observed among GA categories, revealing an aggregated spatial pattern [g (r) > 1] that emphasizes the necessity for tailored strategies to enhance and conserve each GA category within the city. Furthermore, there is a perception of critical degradation in various GA categories, namely city bush, cropland, and forest plantation. The primary causes identified for GA depletion in the city were poor management strategies and lack of planning. These results could provide valuable guidance for policymakers, urban planners, and cityscape architects with a focus on urban sustainability, particularly regarding the development of GAs in the Republic of Benin.

    Citation: Bokon A Akakpo, Elie A Padonou, Appollonia A Okhimamhe, Emmanuel T Umaru, Akomian F Azihou, Haruna Ibrahim, Vincent AO Orekan, Brice A Sinsin. Spatio-temporal alterations, configurations, and distribution of green areas, along with their sustainability in Parakou, Benin[J]. AIMS Geosciences, 2024, 10(3): 553-572. doi: 10.3934/geosci.2024029

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  • Green areas (GAs) are swiftly declining in urban areas worldwide, amplifying adverse local climate impacts on the well-being of city residents. Despite this, there is limited empirical research on the changing patterns and distribution of GAs and their vulnerability. This is especially notable in dry tropical cities where these spaces function as vital microclimate areas that control against climate change effects such as flooding and heat islands. This study focused on examining the changing GA coverage, scrutinizing the spatial distribution of different GA categories, and investigating threat factors associated with their perceived sustainability in Parakou. Employing a mixed-methods approach, open-source geospatial data and collected primary data were acquired through on-site observations as well as semi-structured interviews. Data analysis involved the application of geospatial, statistical, and textual techniques. The results indicated that, from 2000 to 2020, the city experienced a loss of 16.48 km2 (24.73%) in its GA cover. The predominant land use change observed was the conversion of sparse vegetation (21.86%) into built-up areas. A notable difference (P < 0.0001) was observed among GA categories, revealing an aggregated spatial pattern [g (r) > 1] that emphasizes the necessity for tailored strategies to enhance and conserve each GA category within the city. Furthermore, there is a perception of critical degradation in various GA categories, namely city bush, cropland, and forest plantation. The primary causes identified for GA depletion in the city were poor management strategies and lack of planning. These results could provide valuable guidance for policymakers, urban planners, and cityscape architects with a focus on urban sustainability, particularly regarding the development of GAs in the Republic of Benin.



    Let Z be a random variable (RV) having probability density function (PDF) gZ(z). Shannon [33] defined entropy for a RV Z as

    H(Z)=0gZ(z)loggZ(z)dz.

    The non-additive generalization of Shannon's entropy of order η, suggested by Tsallis [37], is known as Tsallis entropy. This measure plays an important role in the uncertainty measurements of an RV Z, which is defined as

    Hη(Z)=1η1(10gηZ(z)dz), (1.1)

    where 0<η1. When η1, Tsallis entropy approaches Shannon entropy.

    There are many applications of this new entropy, especially in physics [7], earthquakes [2], stock exchanges [20], plasma [23], and income distribution [35]. For more information about Tsallis entropy, we recommend reading Tsallis [38]. Several generalizations of Shannon entropy have been developed, which make these entropies sensitive to different kinds of probability distributions via the addition of a few additional parameters. A new measure of Shannon entropy, cumulative residual entropy (CRE), was introduced by Rao et al. [30] by taking into account the survival function instead of the probability density function. CRE is considered more stable and mathematically sound due to its more regular survival function (SF) than the PDF. Moreover, distribution functions exist even when probability density functions do not exist (e.g., Govindarajulu, power-Pareto, and generalized lambda distributions). CRE measure is based on SF ¯GZ(z). According to his definition, CRE is defined as

    J(Z)=0¯GZ(z)log¯GZ(z)dz.

    A cumulative residual Tsallis entropy (CRTE) of order η, which is represented by ζη(Z), was introduced by Sati and Gupta [32]. This CRTE is defined as

    ζη(Z)=1η1(10¯GηZ(z)dz),η>0,η1. (1.2)

    When η1, CRTE approaches CRE.

    The CRTE may also be represented in terms of the mean residual life function of Z, which is another useful representation defined as

    ζη(Z)=1ηE[m(Zη)]. (1.3)

    Rajesh and Sunoj [31] unveiled an alternative measure for CRTE denoted by the order η, which is defined as

    ξη(Z)=1η1(0(¯GZ(z)¯GηZ(z))dz),η>0,η1. (1.4)

    The characteristics of the residual Tsallis entropy for order statistics (OSs) were studied by Shrahili and Kayid [34]. Mohamed [24] recently conducted a study on the CRTE and its dynamic form, which is based on the Farlie-Gumbel-Morgenstern (FGM) family. When prior information is presented in the form of marginal distributions, it is advantageous to model bivariate data using marginal distributions. The FGM family is one of these families that has been the subject of a significant amount of study. The FGM family is represented by the bivariate cumulative distribution function (CDF) GZ,X(z,x)=GZ(z)GX(x)[1+θ(1GZ(z))(1GX(x))], 1θ1, where GZ(z) and GX(x) are the marginal CDFs of two RVs Z and X, respectively. Literature indicates that several modifications have been implemented in the FGM family to increase the correlation between its marginals. Extensive families have been the subject of a great number of studies, each of which has a unique point of view. Examples of these studies are Barakat et al. [6], Abd Elgawad and Alawady [1], Alawady et al. [4], Chacko and Mary [10], Husseiny et al. [17,18], and Nagy et al. [26]. It was demonstrated by Huang and Kotz [15] that a single iteration may result in a doubling of the correlation between marginals in FGM. This was established through the use of a single iteration. The joint CDF iterated FGM (IFGM) family with a single iteration is denoted by IFGM(γ,ω) and defined as

    GZ,X(z,x)=GZ(z)GX(x)[1+γ¯GZ(z)¯GX(x)+ωGZ(z)GX(x)¯GZ(z)¯GX(x)]. (1.5)

    The corresponding joint PDF (JPDF) is given by

    gZ,X(z,x)=gZ(z)gX(x)[1+γ(12GZ(z))(12GX(x))+ωGZ(z)GX(x)(23GZ(z))(23GX(x))]. (1.6)

    Classical FGM can clearly be regarded as a special case of the IFGM(γ,ω) family (1.5)–(1.6) by putting ω=0. If the two marginals GZ(z) and GX(x) are continuous, Huang and Kotz [15] showed that the natural parameter space Ω (which is the admissible set of the parameters γ and ω that makes GZ,X(z,x) is a genuine CDF) is convex, where Ω={(γ,ω):1γ1;1γ+ω;ω3γ+96γ3γ22}. Additionally, if the marginals are uniform, the correlation coefficient is ρ=γ3+ω12. Finally, the maximal correlation coefficient attained for this family is maxρ=0.434, versus maxρ=13=0.333 achieved for γ=1 in the original FGM [16]. The JPDF of the IFGM copula is plotted in Figure 1. Figure 1 illustrates subfigures that exhibit unique parameter values. Each subfigure from (a) to (f) had the parameter values arranged in a vector form (γ,ω).

    Figure 1.  The JPDF for IFGM copula.

    As a unifying model for ascendingly ordered RVs, generalized order statistics (GOSs) have drawn more and more attention. The GOSs model was first presented by Kamps [21]. It is made up of several pertinent models of ordered RVs, such as order statistics (OSs), record values, sequential OSs (SOSs), and progressive censored type-Ⅱ OSs (POS-II). The RVs Z(r,n,˜m,κ),r=1,2,...,n, are called GOSs based on a continuous CDF GZ(z) with the PDF gZ(z), if their JPDF has the form

    f(˜m,κ)1,...,n:n(z1,...,zn)=κGγn1Z(zn)gZ(zn)n1i=1γiGγiγi+11Z(zi)gZ(zi),

    where G1(0)z1...znG1(1), κ>0,γi=n+κi+n1t=imt>0,i=1,,n1, and ˜m=(m1,m2,,mn1)R. In this paper, we assume that the parameters γ1,,γn1, and γn=κ, are pairwise different, i.e., γtγs,ts,t,s=1,2,...,n. We obtain a very wide subclass of GOSs that contains m-GOSs (where m1=...=mn1=m), OSs, POS-II, and SOSs. The PDF of the rth GOS and the JPDF of the rth and sth GOSs, 1r<sn, respectively, are given by Kamps and Cramer [22].

    fZ(r,n,˜m,κ)(z)=Crri=1αi;r¯Gγi1Z(z)gZ(z),zR,1rn, (1.7)
    fZ(r,n;˜m,κ),Z(s,n;˜m,κ)(z,x)=Cs[si=r+1αi;r;s(¯GZ(x)¯GZ(z))γi][ri=1αi;r¯GγiZ(z)]gZ(z)¯GZ(z)gZ(x)¯GZ(x),z<x, (1.8)

    where ¯G=1G is (SF) of G, Cr=ri=1γi,αi;r=rj=1ji1γjγi,1irn, and αi;r;s=sj=r+1ji1γjγi,r+1isn.

    When dealing with selection and prediction difficulties, the meaning of concomitants is a vital tool. The idea of concomitants of OSs (COSs) was first proposed by David [11]. Refer to David and Nagaraja [12] for a comprehensive understanding of the COS. Many studies have been published on the concomitants of the GOSs (CGOSs) model. Researchers such as Alawady et al. [5], Beg and Ahsanullah [8], and Domma and Giordano [13] have studied this issue. The CGOSs models, however, have only been studied in a restricted number of studies when γtγs,ts,t,s=1,2,...,n. These include Abd Elgawad and Alawady [1], and Mohie El-Din et al. [25].

    Let (Zi,Xi),i=1,2,...,n, be a random sample from a continuous bivariate CDF GZ,X(z,x). If we denote Z(r,n,˜m,κ) as the rth GOS of the Z sample values, then the X values associated with Z(r,n,˜m,κ) is called the concomitant of the rth GOS and is denoted by X[r,n,˜m,κ],r=1,2,...,n. The PDF of the concomitant of rth GOS is given by

    g[r,n,˜m,κ](x)=gX|Z(x|z)fZ(r,n,˜m,κ)(z)dz. (1.9)

    More generally, for 1r<sn, the JPDF of the concomitants of rth and sth GOSs is given by

    g[r,s,n,˜m,κ](x1,x2)=x1gX|Z(x1|z1)gX|Z(x2|z2)fZ(r,n,˜m,κ),Z(s,n,˜m,κ)(z1,z2)dz2dz1. (1.10)

    Motivation and the purpose of the work

    Mohamed [24] exhibited CRTE features in CGOSs that were based on FGM. Suter et al. [36] conducted another study that examined Tsallis entropy in CGOSs resulting from FGM. We generalize the previous articles by investigating Tsallis measures in CGOS from IFGM in more general scenarios. The objectives that inspired this study are as follows: Tsallis entropy measures based on CGOSs with interesting features are introduced in a broad framework. We considered sub models through a comprehensive numerical analysis, including OSs, record values, and k-record values. An in-depth analysis of reaching satisfactory results using the nonparametric estimate of these measures. More sophistication and flexibility are provided by the suggested distribution (IFGM) for modeling complicated data sets. This is why we used actual data in our analysis.

    The arrangement of this paper is organized as follows: In Section 2, we obtain some characterization results on concomitants X[r,n,˜m,k] based on IFGM(γ,ω) as Tsallis entropy, CRTE, and alternate measure of CRTE. In Section 3, we extend and compute some examples of information measures for the concomitants X[r,n,˜m,k] from IFGM(γ,ω). We use the empirical method in combination with CGOS based on the IFGM family, to estimate the CRTE in Section 4. Finally, in Section 5, a bivariate real-world data set has been probed, and we examine the Tsallis entropy and CRTE. Finally, Section 6 concludes the work.

    In this section, we derived Tsallis entropy, CRTE, and an alternative measure CRTE for CGOS based on the IFGM(γ,ω) family. First, we will point out some important results that we will use in deducing these measures. Husseiny et al. [17] derived the PDF, CDF, and SF for the concomitant X[r,n,˜m,k] of the rth GOS, respectively, as follows:

    g[r,n,˜m,k](x)=(1+δ(˜m,k)r,n:1)gX(x)+(δ(˜m,k)r,n:2δ(˜m,k)r,n:1)gV1(x)δ(˜m,k)r,n:2gV2(x), (2.1)
    G[r,n,˜m,k](x)=GX(x)[1+δ(˜m,k)r,n:1(1GX(x))+δ(˜m,k)r,n:2(GX(x)G2X(x))], (2.2)

    and

    ¯G[r,n,˜m,k](x)=¯GX(x)[1δ(˜m,k)r,n:1GX(x)δ(˜m,k)r,n:2G2X(x)], (2.3)

    where ViGi+1X,i=1,2, δ(˜m,k)r,n:1=γCr1ri=1ai(r)(1γi1+γi) and δ(˜m,k)r,n:2=ωCr1ri=1ai(r)(1γi1+γi)(3γi2+γi).

    Theorem 2.1. Tsallis entropy of concomitants of the rth GOS based on the IFGM(γ,ω) is given by

    Hη[r,n,˜m,k](x)=1η1(1N(η)j=0jp=0(ηj)(jp)(δ(˜m,k)r,n:1)jp(δ(˜m,k)r,n:2)pEU[(gX(G1X(U)))η1(12U)jp(2U3U2)p]),

    where N(x)=, if x is non-integer, and N(x)=x, if x is integer, and U is a uniform RV on (0, 1).

    Proof. Using (1.1) and (2.1), Tsallis entropy is provided by

    Hη[r,n,˜m,k](x)=1η1(10gη[r,n,˜m,k](x)dx)=1η1(10gηX(x)(1+δ(˜m,k)r,n:1(12GX(x))+δ(˜m,k)r,n:2GX(x)(23GX(x)))ηdx)=1η1(1N(η)j=0jp=0(ηj)(jp)(δ(˜m,k)r,n:1)jp(δ(˜m,k)r,n:2)pE[gη1X(x)(12GX(x))jp(2GX(x)3GX(x)2)p]). (2.4)

    Remark 2.1. If ˜m=0 and k=1. The Tsallis entropy of the concomitant of the rth OS based on the IFGM(γ,ω) is given by

    Hη[r:n](x)=1η1(1N(η)j=0jp=0(ηj)(jp)(Ω1,r:n)jp(Ω2,r:n)pE[gη1X(x)(12GX(x))jp(2GX(x)3GX(x)2)p])

    where Ω1,r:n=γ(n2r+1)n+1 and Ω2,r:n=ω[r(2n3r+1)(n+1)(n+2)], (cf. Husseiny et al. [17]).

    Remark 2.2. If ˜m=1 and k=1. Tsallis entropy of the concomitant of the nth upper record value based on IFGM(γ,ω) is given by

    Hη[n](x)=1η1(1N(η)j=0jp=0(ηj)(jp)(Δn:1)jp(Δn:2)pE[gη1X(x)(12GX(x))jp(2GX(x)3GX(x)2)p]),

    where Δn:1=γ(2(n1)1) and Δn:2=ω(2(n2)3(n1)1).

    Remark 2.3. Tsallis entropy for the concomitant of the nth upper k-record value based on IFGM(γ,ω) is given by

    Hη[n,k](x)=1η1(1N(η)j=0jp=0(ηj)(jp)(n,k:1)jp(n,k:2)pE[gη1X(x)(12GX(x))jp(2GX(x)3GX(x)2)p]),

    where n,k:1=γ(2(kk+1)n1) and n,k:2=ω(4(kk+1)n3(kk+2)n1). (cf. Nagy and Alrasheedi [27]).

    Theorem 2.2. CRTE for CGOS based on the IFGM(γ,ω) is given by

    ζη[r,n,˜m,k](x)=1η1(1N(η)i=0is=0(ηi)(is)(1)2is(δ(˜m,k)r,n:1)s(δ(˜m,k)r,n:2)isEU[(U)2is(1U)ηgX(G1X(U))]),

    where U is a uniform RV on (0, 1).

    Proof. Using (1.2) and (2.3), then CRTE is provided by

    ζη[r,n,˜m,k](x)=1η1(10¯Gη[r,n,˜m,k](x)dx)=1η1(10¯GηX(x)[1GX(x)(δ(˜m,k)r,n:1δ(˜m,k)r,n:2GX(x))]ηdx)=1η1(1N(η)i=0is=0(ηi)(is)(1)2is(δ(˜m,k)r,n:1)s(δ(˜m,k)r,n:2)isE[g1X(x)(1GX(x))η(GX(x))2is]). (2.5)

    Remark 2.4. If ˜m=0 and k=1. The CRTE of the concomitant of the rth OS based on the IFGM(γ,ω) is given by

    ζη[r:n](x)=1η1(1N(η)i=0is=0(ηi)(is)(1)2is(Ω1,r:n)s(Ω2,r:n)isE[g1X(x)(1GX(x))η(GX(x))2is]).

    Remark 2.5. If ˜m=1 and k=1. CRTE of the concomitant of the nth upper record value based on the IFGM(γ,ω) is given by

    ζη[n](x)=1η1(1N(η)i=0is=0(ηi)(is)(1)2is(Δn:1)s(Δn:2)isE[g1X(x)(1GX(x))η(GX(x))2is]).

    Remark 2.6. CRTE of the concomitant of the nth upper k-record value based on the IFGM(γ,ω) is given by

    ζη[n,k](x)=1η1(1N(η)i=0is=0(ηi)(is)(1)2is(n,k:1)s(n,k:2)isE[g1X(x)(1GX(x))η(GX(x))2is]).

    For the concomitant X[r,n,˜m,k] of the rth GOS, the moment of X[r,n,˜m,k] based on the IFGM(γ,ω) (cf. Husseiny et al. [17]) is given by

    μ[r,n,˜m,k](x)=(1+δ(˜m,k)r,n:1)μX+(δ(˜m,k)r,n:2δ(˜m,k)r,n:1)μV1δ(˜m,k)r,n:2μV2. (2.6)

    Theorem 2.3. The alternative measure of CRTE for CGOS based on IFGM(γ,ω) is given by

    ξη[r,n,˜m,k](x)=1η1(μ[r,n,˜m,k](x)N(η)i=0is=0(ηi)(is)(1)2is(δ(˜m,k)r,n:1)s(δ(˜m,k)r,n:2)isEU[(U)2is(1U)ηgX(G1X(U))]),

    where U is a uniform RV on (0, 1).

    Proof. Using (1.4) and (2.6), the alternative measure of CRTE is provided by

    ξη[r,n,˜m,k](x)=1η1(0(¯G[r,n,˜m,k](x)¯Gη[r,n,˜m,k](x))dx)=1η1(μ[r,n,˜m,k](x)0¯GηX(x)[1GX(x)(δ(˜m,k)r,n:1δ(˜m,k)r,n:2GX(x))]ηdx)=1η1(μ[r,n,˜m,k](x)N(η)i=0is=0(ηi)(sp)(1)2is(δ(˜m,k)r,n:1)s(δ(˜m,k)r,n:2)isE[g1X(x)(1GX(x))η(GX(x))2is]). (2.7)

    Theorem 2.4. Let X[r,n,˜m,k] be a CGOS based on a continuous CDF GX(x) with the PDF gX(x). For all η>0, we have

    ξη[r,n,˜m,k](x)=E(Xη[r,n,˜m,k])+E(Hη[r,n,˜m,k](X)),

    where

    Hη[r,n,˜m,k](u)=u0m[r,n,˜m,k](x)¯Gη1[r,n,˜m,k](x)dx,u>0.

    Proof. Using (1.3) and m[r,n,˜m,k](x)λ[r,n,˜m,k](x)=1+m[r,n,˜m,k](x), where

    λ[r,n,˜m,k](x)=g[r,n,˜m,k](x)¯G[r,n,˜m,k](x).

    Then, we obtain

    ξη[r,n,˜m,k](x)=0m[r,n,˜m,k](x)λ[r,n,˜m,k](x)¯Gη[r,n,˜m,k](x)dx=E(Xη[r,n,˜m,k])+0m[r,n,˜m,k](x)¯Gη[r,n,˜m,k](x)dx,

    for all η>0. Upon using Fubini's theorem, we obtain

    0m[r,n,˜m,k](x)¯Gη[r,n,˜m,k](x)dx=0m[r,n,˜m,k](x)dxxg[r,n,˜m,k](u)¯Gη1[r,n,˜m,k](x)dudx=0g[r,n,˜m,k](u)u0m[r,n,˜m,k](x)¯Gη1[r,n,˜m,k](x)dxdu.

    This gives the desired result.

    Remark 2.7. If ˜m=0 and k=1. The alternative measure of CRTE for concomitant of the rth OSs based on the IFGM(γ,ω) is given by

    ξη[r:n](x)=1η1(μ[r:n](x)N(η)i=0is=0(ηi)(is)(1)2is(Ω1,r:n)s(Ω2,r:n)isE[g1X(x)(1GX(x))η(GX(x))2is]).

    Remark 2.8. If ˜m=1 and k=1. The alternative measure of CRTE for concomitant of the nth upper record value based on the IFGM(γ,ω) is given by

    ξη[n](x)=1η1(μ[n](x)N(η)i=0is=0(ηi)(is)(1)2is(Δn:1)s(Δn:2)isE[g1X(x)(1GX(x))η(GX(x))2is]).

    Remark 2.9. The alternative measure of CRTE for concomitant of the nth upper k-record value based on the IFGM(γ,ω) is given by

    ξη[n,k](x)=1η1(μ[n,k](x)N(η)i=0is=0(ηi)(is)(1)2is(n,k:1)s(n,k:2)isE[g1X(x)(1GX(x))η(GX(x))2is]).

    In this section, we study the Tsallis entropy, CRTE, and alternate measure of CRTE for CGOS in IFGM(γ,ω) for some popular distributions. We consider the extended Weibull (EW) family of distributions, which was developed by Gurvich et al. [14], as a case study. The CDF of EW is given by

    GX(x)=1eτH(x;ε),x>0,τ>0,

    where H(x;ε) is a differentiable, nonnegative, continuous, and monotonically increasing function when x depends on the parameter vector ε. Also, H(x;ε)0+ as x0+ and H(x;ε)+ as x+. This CDF is denoted by EW (τ,ε) and has the following PDF:

    gX(x)=τh(x;ε)eτH(x;ε),x>0,

    where h(x;ε) is the derivative of H(x;ε) with respect to x. Several important models are included in the EW, including the Rayleigh, Pareto, Weibull, uniform, and exponential distributions (ED). For further details about this family, see Jafari et al. [19].

    Example 3.1. Consider two variables, Z and X, that possess ED from IFGM (represented by IFGM-ED) (i.e. GX(x)=1eθx,x,θ>0). Based on (2.4), we get the Tsallis entropy in X[r,n,˜m,k] as follows:

    Hη[r,n,˜m,k](x)=1η1(1N(η)j=0jp=0jpl=0pu=0(ηj)(jp)(jpl)(pu)(1)l+u(2)jpl(3)pu(δ(˜m,k)r,n:1)jp(δ(˜m,k)r,n:2)pθη1β(1+p,η+ju1)).

    Example 3.2. Consider Z and X to be power distributions derived from IFGM (i.e. GX(x)=xc,0x1,c>0). Then Hη[r,n,˜m,k](x) is given by

    Hη[r,n,˜m,k](x)=1η1(1N(η)j=0jp=0jpt=0pw=0(ηj)(jp)(jpt)(pw)(1)t(2)t+pw(3)w(δ(˜m,k)r,n:1)jp(δ(˜m,k)r,n:2)pcη1η+c(t+p+w+η)).

    Example 3.3. Suppose that Z and X have EW based on IFGM with (i.e. GX(x)=1eτH(x;ε),x>0,τ>0,). Then, we have the Tsallis entropy in X[r,n,˜m,k] as follows:

    H(EW)η[r,n,˜m,k](x)=1η1(1N(η)j=0jp=0(ηj)(jp)(δ(˜m,k)r,n:1)jp(δ(˜m,k)r,n:2)pE[[(τh(x;ε))e(τH(x;ε))]η1(2e(τH(x;ε))1)jp(4e(τH(x;ε))3(e(τH(x;ε)))21)p]).

    Example 3.4. Assume that Z and X both possess IFGM-ED. Based on (2.5), we obtain the CRTE in X[r,n,˜m,k] as follows:

    ζη[r,n,˜m,k](x)=1θ(η1)(1N(η)i=0is=0(ηi)(is)(1)2is(δ(˜m,k)r,n:1)s(δ(˜m,k)r,n:2)isβ(2is+1,η)).

    Example 3.5. Assume that the uniform distributions of Z and X come from an IFGM (i.e. GX(x)=x,0x1). Based on (2.5), we obtain the CRTE in X[r,n,˜m,k] as follows:

    ζη[r,n,˜m,k](x)=1η1(1N(η)i=0is=0(ηi)(is)(1)2is(δ(˜m,k)r,n:1)s(δ(˜m,k)r,n:2)isβ(2is+1,η+1)).

    Example 3.6. Let us say that Z and X have EW according to IFGM. From (2.5), we get the CRTE in X[r,n,˜m,k] as follows:

    ζ(EW)η[r,n,˜m,k](x)=1η1(1N(η)i=0is=0(ηi)(is)(1)2is(δ(˜m,k)r,n:1)s(δ(˜m,k)r,n:2)isE[(τh(x;ε)eτH(x;ε))1(eτH(x;ε))η(1eτH(x;ε))2is]).

    Example 3.7. Assume that Z and X both possess IFGM-ED. Based on (2.7), we have the alternate measure of CRTE in X[r,n,˜m,k] as follows:

    ξη[r,n,˜m,k](x)=1θ(η1)((1δ(˜m,k)r,n:12δ(˜m,k)r,n:23)N(η)i=0is=0(ηi)(is)(1)2is(δ(˜m,k)r,n:1)s(δ(˜m,k)r,n:2)isβ(2is+1,η)).

    Example 3.8. Assume that the uniform distributions of Z and X come from IFGM. Based on (2.7), we obtain the alternate measure of CRTE in X[r,n,˜m,k] as follows:

    ξη[r,n,˜m,k](x)=1(η1)(12(1δ(˜m,k)r,n:13δ(˜m,k)r,n:26)N(η)i=0is=0(ηi)(is)(1)2is(δ(˜m,k)r,n:1)s(δ(˜m,k)r,n:2)isβ(2is+1,η+1)).

    Example 3.9. Let us say that Z and X have EW, according to IFGM. Based on (2.7), we obtain the alternate measure of CRTE in X[r,n,˜m,k] as follows:

    ξη[r,n,˜m,k](x)=1η1(μEW[r,n,˜m,k](x)N(η)i=0is=0(ηi)(is)(1)2is(δ(˜m,k)r,n:1)s(δ(˜m,k)r,n:2)isE[(τh(x;ε)eτH(x;ε))1(eτH(x;ε))η(1eτH(x;ε))2is]).

    As shown in Tables 14 of the IFGM-ED, the Tsallis entropy and the CRTE for X[r:n] and X[n] are presented. After running the numbers through MATHEMATICA version 12, we can deduce the following properties from Tables 14.

    Table 1.  Tsallis entropy for X[r:n] based on IFGM-ED.
    γ=0.9,η=5,θ=0.5 γ=0.5,η=10,θ=1
    n r ω=1.8 ω=1.4 ω=1.2 ω=1.4 n r ω=0.4 ω=1.4 ω=2.5 ω=3
    3 1 0.23536 0.23456 0.22795 0.22733 3 1 0.11068 0.10937 0.10798 0.10707
    3 2 0.24673 0.2466 0.24559 0.24551 3 2 0.09946 0.09588 0.09301 0.09149
    3 3 0.24888 0.24907 0.24972 0.24974 3 3 0.00306 0.04804 0.06282 0.06771
    7 1 0.22023 0.21907 0.21008 0.20928 7 1 0.11495 0.11308 0.11223 0.11183
    7 2 0.23621 0.23526 0.22701 0.2262 7 2 0.11074 0.109 0.1068 0.10522
    7 3 0.24371 0.2432 0.23845 0.23797 7 3 0.10794 0.10384 0.09844 0.09471
    7 4 0.2471 0.2469 0.24523 0.24506 7 4 0.09989 0.09348 0.08732 0.08372
    7 5 0.24852 0.24852 0.24852 0.24852 7 5 0.0751 0.0751 0.0751 0.0751
    7 6 0.24902 0.24916 0.24969 0.24971 7 6 0.00542 0.04396 0.05796 0.06282
    7 7 0.24903 0.24935 0.24993 0.24993 7 7 -0.17228 -0.0084 0.02783 0.03836
    9 1 0.21557 0.21442 0.20583 0.20508 9 1 0.1167 0.11456 0.11364 0.11326
    9 2 0.23139 0.23026 0.22071 0.21979 9 2 0.11187 0.11031 0.10888 0.1079
    9 3 0.24007 0.23928 0.23201 0.23126 9 3 0.10984 0.10723 0.10341 0.10059
    9 4 0.24474 0.24428 0.24 0.23955 9 4 0.10703 0.10192 0.09523 0.09064
    9 5 0.24717 0.24696 0.24514 0.24496 9 5 0.09999 0.09288 0.08576 0.08151
    9 6 0.24838 0.24834 0.24803 0.24801 9 6 0.0826 0.07875 0.07607 0.07475
    9 7 0.24893 0.24901 0.24938 0.2494 9 7 0.04208 0.05741 0.06448 0.06724
    9 8 0.2491 0.24929 0.24985 0.24986 9 8 -0.04679 0.02525 0.04694 0.05386
    9 9 0.24901 0.24937 0.24994 0.24993 9 9 -0.22959 -0.02309 0.01896 0.03111

     | Show Table
    DownLoad: CSV
    Table 2.  Tsallis entropy for X[n] based on IFGM-ED.
    γ=0.9,η=5,θ=0.5 γ=0.5,η=10,θ=1
    n ω=1.8 ω=1.4 ω=1.2 ω=1.4 n ω=0.4 ω=1.4 ω=2.5 ω=3
    2 0.24877 0.249 0.24973 0.24976 2 0.00145 0.05054 0.06565 0.07051
    3 0.24895 0.24931 0.24993 0.24993 3 -0.1754 -0.00544 0.03056 0.04099
    4 0.24888 0.24934 0.24994 0.24993 4 -0.33831 -0.04564 0.00606 0.0216
    5 0.24878 0.24933 0.24994 0.24992 5 -0.44823 -0.06915 -0.00781 0.01218
    6 0.24872 0.24932 0.24993 0.24992 6 -0.51214 -0.0817 -0.01496 0.00829
    7 0.24868 0.24931 0.24993 0.24992 7 -0.54667 -0.08814 -0.01851 0.00677
    8 0.24866 0.2493 0.24993 0.24991 8 -0.56464 -0.09139 -0.02027 0.00617
    9 0.24865 0.2493 0.24993 0.24991 9 -0.57382 -0.09301 -0.02113 0.00593
    10 0.24864 0.2493 0.24993 0.24991 10 -0.57846 -0.09382 -0.02156 0.00582
    11 0.24864 0.2493 0.24993 0.24991 11 -0.58079 -0.09423 -0.02177 0.00577
    12 0.24864 0.2493 0.24993 0.24991 12 -0.58196 -0.09443 -0.02188 0.00575
    13 0.24864 0.2493 0.24993 0.24991 13 -0.58255 -0.09453 -0.02193 0.00574
    14 0.24864 0.2493 0.24993 0.24991 14 -0.58284 -0.09458 -0.02196 0.00573
    15 0.24864 0.2493 0.24993 0.24991 15 -0.58299 -0.09461 -0.02197 0.00573
    16 0.24864 0.2493 0.24993 0.24991 16 -0.58306 -0.09462 -0.02198 0.00573
    17 0.24864 0.2493 0.24993 0.24991 17 -0.5831 -0.09463 -0.02198 0.00573
    18 0.24864 0.2493 0.24993 0.24991 18 -0.58312 -0.09463 -0.02198 0.00573
    19 0.24864 0.2493 0.24993 0.24991 19 -0.58313 -0.09463 -0.02198 0.00573
    20 0.24864 0.2493 0.24993 0.24991 20 -0.58313 -0.09463 -0.02198 0.00573
    21 0.24864 0.2493 0.24993 0.24991 21 -0.58314 -0.09463 -0.02198 0.00573

     | Show Table
    DownLoad: CSV
    Table 3.  CRTE for X[r:n] based on IFGM-ED.
    γ=0.9,η=5,θ=0.5 γ=0.5,η=10,θ=1
    n r ω=1.8 ω=1.4 ω=1.2 ω=1.4 n r ω=0.4 ω=1.4 ω=2.5 ω=3
    3 1 0.43317 0.43238 0.42624 0.42568 3 1 0.09714 0.09603 0.0952 0.09477
    3 2 0.40404 0.40318 0.39702 0.39651 3 2 0.10007 0.09975 0.09955 0.09945
    3 3 0.32162 0.32909 0.36567 0.3678 3 3 0.10199 0.10251 0.10277 0.10289
    7 1 0.44158 0.4411 0.4375 0.43719 7 1 0.09509 0.09389 0.09302 0.09258
    7 2 0.43401 0.43307 0.42539 0.42465 7 2 0.0972 0.09578 0.09466 0.09406
    7 3 0.42314 0.42183 0.41092 0.40986 7 3 0.09882 0.09775 0.0969 0.09645
    7 4 0.40648 0.40515 0.3949 0.39398 7 4 0.10011 0.09958 0.0992 0.09901
    7 5 0.37863 0.37863 0.37863 0.37863 7 5 0.10115 0.10115 0.10115 0.10115
    7 6 0.32728 0.33319 0.36347 0.36531 7 6 0.10201 0.10245 0.10267 0.10277
    7 7 0.22427 0.25007 0.35054 0.35503 7 7 0.10273 0.10352 0.10409 0.10451
    9 1 0.44288 0.44248 0.43956 0.43931 9 1 0.09461 0.09351 0.09273 0.09234
    9 2 0.43748 0.4367 0.4304 0.4298 9 2 0.09643 0.09495 0.09378 0.09317
    9 3 0.43039 0.42925 0.41959 0.41864 9 3 0.09791 0.09651 0.09538 0.09476
    9 4 0.42075 0.41932 0.40741 0.40626 9 4 0.09911 0.09808 0.09725 0.09682
    9 5 0.40701 0.40558 0.3944 0.39339 9 5 0.10012 0.09954 0.09911 0.0989
    9 6 0.3863 0.38566 0.38127 0.38092 9 6 0.10097 0.10083 0.10074 0.1007
    9 7 0.353 0.35534 0.36876 0.36968 9 7 0.10169 0.10194 0.10208 0.10214
    9 8 0.29611 0.30699 0.35753 0.3603 9 8 0.10232 0.10288 0.10318 0.10332
    9 9 0.1942 0.22695 0.34802 0.35309 9 9 0.10286 0.10373 0.10449 0.10513

     | Show Table
    DownLoad: CSV
    Table 4.  CRTE for X[n] based on IFGM-ED.
    γ=0.9,η=5,θ=0.5 γ=0.5,η=10,θ=1
    n ω=1.8 ω=1.4 ω=1.2 ω=1.4 n ω=0.4 ω=1.4 ω=2.5 ω=3
    2 0.31766 0.32626 0.3671 0.3694 2 0.10198 0.10254 0.10284 0.10297
    3 0.21824 0.24603 0.35185 0.35646 3 0.10272 0.10355 0.10418 0.10466
    4 0.13412 0.18248 0.34541 0.35144 4 0.10305 0.10408 0.10545 0.10682
    5 0.07755 0.14133 0.34267 0.34947 5 0.1032 0.10439 0.10658 0.10897
    6 0.04423 0.11758 0.34147 0.34867 6 0.10327 0.10456 0.10737 0.11053
    7 0.02598 0.10473 0.34094 0.34833 7 0.10331 0.10466 0.10785 0.11149
    8 0.01638 0.098 0.34069 0.34818 8 0.10332 0.10471 0.10812 0.11203
    9 0.01144 0.09456 0.34057 0.34812 9 0.10333 0.10474 0.10826 0.11231
    10 0.00893 0.09281 0.34052 0.34809 10 0.10334 0.10475 0.10833 0.11246
    11 0.00766 0.09193 0.34049 0.34807 11 0.10334 0.10476 0.10837 0.11253
    12 0.00702 0.09148 0.34048 0.34806 12 0.10334 0.10476 0.10838 0.11257
    13 0.0067 0.09126 0.34047 0.34806 13 0.10334 0.10477 0.10839 0.11259
    14 0.00654 0.09115 0.34047 0.34806 14 0.10334 0.10477 0.1084 0.1126
    15 0.00646 0.09109 0.34046 0.34806 15 0.10334 0.10477 0.1084 0.1126
    16 0.00642 0.09107 0.34046 0.34806 16 0.10334 0.10477 0.1084 0.11261
    17 0.0064 0.09105 0.34046 0.34806 17 0.10334 0.10477 0.1084 0.11261
    18 0.00639 0.09105 0.34046 0.34806 18 0.10334 0.10477 0.1084 0.11261
    19 0.00639 0.09104 0.34046 0.34806 19 0.10334 0.10477 0.1084 0.11261
    20 0.00639 0.09104 0.34046 0.34806 20 0.10334 0.10477 0.1084 0.11261
    21 0.00639 0.09104 0.34046 0.34806 21 0.10334 0.10477 0.1084 0.11261
    22 0.00638 0.09104 0.34046 0.34806 22 0.10334 0.10477 0.1084 0.11261

     | Show Table
    DownLoad: CSV

    ● When γ=0.9, η=5, and θ=0.5, the value of Hη[r:n](x) goes up as n goes up. When γ=0.5, η=10, and θ=1, the value of Hη[r:n](x) goes down as n goes up. But Hη[r:n](x) stays the same for all ω values when n=7 and r=5 (look at Table 1).

    ● We see that when γ is 0.9, η is 5, and θ is 0.5, and when γ is -0.5, η is 10, and θ is 1, the value of Hη[n](x) goes down as n goes up, and it almost stays the same when n=10 (look at Table 2).

    ● When γ=0.9, η=5, and θ=0.5, the value of ζη[r:n](x) goes down as n goes up. On the other hand, when γ=0.5, η=10, and θ=1, the value of ζη[r:n](x) goes up as n goes up. It gets bigger as n gets bigger, but ζη[r:n](x) stays the same for all ω values when n=7 and r=5 (look at Table 3).

    ● When γ=0.9, η=5, and θ=0.5, the value of ζη[n](x) goes down as n goes up. When γ=0.5, η=10, and θ=1, the value of ζη[n](x) goes up as n goes up, and the value of ζη[n](x) stays the same at n=22 (look at Table 4).

    For the purpose of calculating the CRTE for concomitant X[r,n,˜m,k], we employ empirical estimators in this section. Next, we'll examine the issue of estimating the CRTE for CGOS using the empirical CRTE. Consider the IFGM sequence (Zi,Xi) for each i=1,2,...,n. In accordance with (2.7), the emperical CRTE of the set X[r,n,˜m,k] can be computed as follows:

    ˆξη[r,n,˜m,k](x)=1η1(0(ˆ¯G[r,n,˜m,k](x)ˆ¯Gη[r,n,˜m,k](x))dx)=1η1(0((1ˆGX(x))[1ˆGX(x)(δ(˜m,k)r,n:1δ(˜m,k)r,n:2ˆGX(x))](1ˆGX(x))η[1ˆGX(x)(δ(˜m,k)r,n:1δ(˜m,k)r,n:2ˆGX(x))]η)dx)=1η1n1j=1x(j+1)x(j)((1ˆGX(x))[1ˆGX(x)(δ(˜m,k)r,n:1δ(˜m,k)r,n:2ˆGX(x))](1ˆGX(x))η[1ˆGX(x)(δ(˜m,k)r,n:1δ(˜m,k)r,n:2ˆGX(x))]ηdx)=1η1n1j=1Δj((1jn)[1jn(δ(˜m,k)r,n:1δ(˜m,k)r,n:2jn)](1jn)η[1jn(δ(˜m,k)r,n:1δ(˜m,k)r,n:2jn)]η),

    where for any CDF G(.), the symbol ˆG(.) stands for the empirical CDF of G(.), and Δj=x(j+1)x(j), j=1,2...,n1, are the sample spacings based on ordered random samples of Xj.

    Example 4.1. Define a random sample from the IFGM-ED as (Zi,Xi), where i ranges from 1 to n. The sample spacings, denoted by Δj, are considered to be independent RVs. Furthermore, Δj exhibits the ED with a mean of 1θ(nj), where j ranges from 1 to n1. For additional information, refer to Chandler [9] and Pyke [29]. Then the expected value and variance of the empirical CRTE in X[r] are given by

    E[ˆξη(X[r])]=1θ(η1)n1j=11(nj)((1jn)[1jn(δ(˜m,k)r,n:1δ(˜m,k)r,n:2jn)](1jn)η[1jn(δ(˜m,k)r,n:1δ(˜m,k)r,n:2jn)]η), (4.1)
    Var[ˆξη(X[r])]=1θ2(η1)2n1j=11(nj)2((1jn)[1jn(δ(˜m,k)r,n:1δ(˜m,k)r,n:2jn)](1jn)η[1jn(δ(˜m,k)r,n:1δ(˜m,k)r,n:2jn)]η)2. (4.2)

    Example 4.2. Again, for completeness, we study here the empirical CRTE of the concomitant X[r] of the rth upper record value Zr based on the IFGM copula. In this case, the sample spacings Δj, j=1,2,...,n1, are independent, and each of them has the beta distribution with parameters 1 and n. According to Pyke [29], the expectation and variance of the empirical CRTE of the concomitant X[r] are as follows:

    E[ˆξη(X[r])]=1(η1)(n+1)n1j=1((1jn)[1jn(δ(˜m,k)r,n:1δ(˜m,k)r,n:2jn)](1jn)η[1jn(δ(˜m,k)r,n:1δ(˜m,k)r,n:2jn)]η),
    Var[ˆξη(X[r])]=n(η1)(n+1)2(n+2)n1j=1((1jn)[1jn(δ(˜m,k)r,n:1δ(˜m,k)r,n:2jn)](1jn)η[1jn(δ(˜m,k)r,n:1δ(˜m,k)r,n:2jn)]η)2.

    Figures 2 and 3 illustrate the relationship between CRTE and empirical CRTE in X[r:n] from IFGM-ED (γ,ω), at n=50. Figures 2 and 3 can be used to obtain the following properties:

    Figure 2.  Representation of CRTE and empirical CRTE in X[r:n] based on IFGM-ED for n=50 and η=20.
    Figure 3.  Representation of CRTE and empirical CRTE in X[r:n] based on IFGM-ED(γ,ω=0) for n=50 and η=20.

    (1) When the θ values are increased, the CRTE and the empirical CRTE have values that are practically identical to one another.

    (2) At most ω and γ values, CRTE and empirical CRTE have identical results, particularly when ω=0 for all r values.

    This section includes analyses of a real-world data set. The data set relates to n=50 simulated simple computer series systems consisting of a processor and a memory. The data was gathered and analyzed based on Oliveira et al. [28]. The data set contains n=50 simulated rudimentary computer systems with processors and memory. An operating computer will be able to operate when both parts are working properly (the processors and memory). Assume the system is nearing the end of its lifecycle. The degeneration advances rapidly in a short period of time [3]. In a short time (in hours), the degeneration advances rapidly. In the case of the first component, a deadly shock can destroy either it or the second component at random, due to the system's greater vulnerability to shocks. We fit the ED to the processor lifetime and memory lifetime separately. As an illustration of the data, Figures 4 and 5 provide a basic statistical analysis. The maximum likelihood estimates of the scale parameters (θi),i=1,2, are 1.24079 and 1.08616, γ=0.175473, and ω=2.16024. Table 5 examines the Tsallis entropy and CRTE for IFGM-ED(0.17543, 2.16024). For the concomitants X[r:50], r=1,2,24,25,49,50, i.e., the lower and upper extremes' concomitants, and the central values' concomitants. We observe that the Hη[r:50](x) and ζη[r:50](x) have maximum values at extremes.

    Figure 4.  Some summary plots of the processor lifetime data set.
    Figure 5.  Some summary plots of memory lifetime data set.
    Table 5.  The Tsallis entropy and CRTE of IFGM-ED at γ2=0.175 and ω2=2.160.
    r 1 2 24 25 49 50
    H5[r:50](x) 0.0736041 0.066275 0.0909935 0.100391 0.23817 0.238162
    ζ5[r:50](x) 0.189322 0.188401 0.177055 0.177312 0.192772 0.193385

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    Given its simplicity and adaptability, IFGM surpasses most FGM generalizations, even though its efficiency is similar to some of those generalizations (such as the Huang\textendash Kotz FGM) in that it has a similar range of correlation coefficients. The CDFs used in this work were consistently formed by linearly combining simpler distributions, due to the advantages they offer. Tsallis entropy and its associated measures for concomitant were derived from IFGM, and a numerical analysis was conducted to uncover certain characteristics of these measures based on GOSs. Special cases were also extracted from this study, for example, OSs, record values, and k-record values. Furthermore, non-parametric estimators of CRTE were derived. The outcomes of an empirical examination of the CRTE are distinct. Finally, an illustrative analysis of a bivariate real-world data set was performed, and the proposed method performs exceptionally well. In the future work, some bivariate distribution families will be considered, including the Huang-kotz, Cambanis, and Sarmanov families, as well as various applications of the CRTE in CGOS. Additionally, we will investigate the quantile function based on Tsallis measures from concomitants. Also, for the estimation problem, we will discuss at least two estimation methods for this model: maximum likelihood and Besyain. Further, a Monte Carlo simulation will be conducted to test the estimator's performance against the empirical measure as well as the exact formula presented in this paper.

    I. A. Husseiny: Conceptualization, Writing original draft, Formal analysis, Software, Investigation, Methodology, Supervision; M. Nagy: Validation, Resources, Writing-review & editing, Data curation, Methodology; A. H. Mansi: Writing-review & editing, Investigation; M. A. Alawady: Conceptualization, Formal analysis, Writing original draft, Software, Investigation, Methodology, Supervision. All authors have read and agreed to the published version of the manuscript.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This research was conducted under a project titled "Researchers Supporting Project", funded by King Saud University, Riyadh, Saudi Arabia under grant number (RSPD2024R969).

    The authors are grateful to the editor and anonymous referees for their insightful comments and suggestions, which helped to improve the paper's presentation.

    This research was conducted under a project titled "Researchers Supporting Project", funded by King Saud University, Riyadh, Saudi Arabia under grant number (RSPD2024R969).

    The authors declare no conflict of interest.



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