A Flexible Bivariate Lifetime Model with Upper Bound: Theoretical Development and Lifetime Application

Abstract

This paper introduces the bivariate bounded Gompertz–log-logistic (BBGLL) distribution, a bounded bivariate lifetime model built by coupling two bounded Gompertz–log-logistic marginals through a Clayton copula with an independent dependence parameter. The proposed model effectively describes positively dependent lifetimes within finite support and accommodates increasing, decreasing, and bathtub-shaped hazard rates. Analytical expressions for the survival functions, hazard rate functions, and joint moments are derived, while measures of association such as Kendall’s tau, Spearman’s rho, and tail-dependence coefficients characterize the dependence structure. Parameters are estimated via maximum likelihood, inference functions for margins (IFM), and semi-parametric methods, with performance assessed through Monte Carlo simulations. A real-life data application illustrates the practical relevance of the model, showing that the BBGLL distribution achieves a superior goodness-of-fit relative to existing bivariate alternatives, highlighting its practical usefulness.

1. Introduction

Modeling bivariate lifetime data is a fundamental topic in reliability theory, survival analysis, and biostatistics where understanding the joint behavior of two interdependent lifetimes, such as components in a system, organs in a biological process, or correlated event times is essential for risk assessment and prediction.

To address such contexts, bounded lifetime distributions have attracted increasing attention. The earliest efforts include bounded variants of well-known univariate models such as the beta, Kumaraswamy, and bounded Weibull families [1,2,3]. These families provide flexibility in modeling monotone or non-monotone hazard rate functions but are often limited to univariate or independent frameworks. Extending them to the bivariate setting remains a challenging task due to the complexity of joint dependence and normalization constraints on finite domains.

To overcome these limitations, copula theory provides a powerful and modular framework for constructing multivariate distributions by linking arbitrary marginal distributions through a dependence function. Copulas separate marginal behavior from dependence structure, allowing researchers to combine heterogeneous marginals while maintaining coherent probabilistic dependence.

The connection between copulas and bivariate distributions is formally established through Sklar’s theorem [4], which states that any bivariate cumulative distribution function (CDF) $F_{Y_1,Y_2}\left(y_1,y_2\right)$ with continuous marginals $F_{Y_1}\left(y_1\right)$ and $F_{Y_2}\left(y_2\right)$ can be uniquely represented as

$$F_{Y_1,Y_2}\left(y_1,y_2\right)=C(F_{Y_1}\left(y_1\right),F_{Y_2}\left(y_2\right)),$$

where $C:{[0,1]}^2\to [0,1]$ is a copula function that fully characterizes the dependence between $Y_1$ and $Y_2$. This decomposition, first formalized by [4] and later developed in [5,6,7], allows for the design of complex multivariate models while preserving flexible marginal specifications. Indeed, copulas permit modeling marginal behaviors and dependence separately, ensuring mathematical tractability and interpretability. Such copula-based constructions have been successfully applied in reliability and hydrological modeling [8] as well as in dependent lifetime analysis [9,10]. Within this framework, the Clayton copula, originally introduced by [11], stands out for its ability to capture positive quadrant dependence (PQD) and nonzero lower-tail dependence, properties particularly relevant to reliability data and biomedical contexts, where simultaneous early failures are likely [7,9,12,13,14]. Indeed, the Clayton family, with its ability to capture asymmetry and provide intuitive tail-dependence modeling, provides a solid foundation for building bivariate lifetime models.

Several authors have proposed bounded or truncated extensions of classical lifetime distributions, including generalized log-logistic and log-logistic-based models for reliability analysis [15]. Copula-based approaches have also been applied in bounded risk and survival contexts, such as bivariate generalized exponential families and truncated bivariate forms arising in actuarial and biomedical studies [10,16]. These developments highlight the growing demand for bivariate lifetime models that combine boundedness, dependence flexibility, and interpretable hazard shapes. The proposed BBGLL model contributes to this line of research by offering a new bounded bivariate family with explicit marginals and a tractable copula-driven dependence structure. Building on this theoretical foundation, this paper proposes a flexible bounded bivariate lifetime distribution, called the bivariate bounded Gompertz–log-logistic (BBGLL) model. The BBGLL family is constructed by coupling two bounded Gompertz–log-logistic (BGLL) marginals, each capable of reproducing increasing, decreasing, and bathtub-shaped hazard rate functions, with a Clayton copula possessing an independent dependence parameter. The BGLL marginal is particularly appealing because it combines the exponential growth pattern of the Gompertz component with the heavy-tailed flexibility of the log-logistic form while maintaining a finite upper bound.

In this work, we establish the theoretical properties of the BBGLL model, including joint and conditional hazard functions, tail-dependence coefficients, and moment expressions. We also present estimation procedures based on maximum likelihood, inference functions for margins (IFM), and semi-parametric methods, together with Monte Carlo simulations assessing finite-sample performance. An empirical application to kidney infection recurrence data illustrates the practical utility of the model and demonstrates its superiority, in terms of AIC, BIC, and log-likelihood criteria, compared to some existing bivariate lifetime models.

Overall, the proposed BBGLL model contributes to the growing literature on bounded and dependent reliability models, providing a flexible, interpretable, and computationally tractable framework for describing realistic bounded lifetime behaviors in engineering, biomedical, and environmental systems.

The remainder of the paper is structured as follows. Section 2 introduces the proposed bivariate distribution, together with its construction and analytical properties. Section 3.2 presents the tractable analytical results, focusing on the moment properties of the distribution. In Section 3.3, we develop parameter estimation methods based on the method of moments and maximum likelihood. Section 4 illustrates the practical usefulness of the model through a real-life data application. Concluding remarks are provided in Section 5.

2. Definitions and Properties

Throughout this paper, we denote the Clayton copula dependence parameter by $\theta$, and for convenience in the marginal BGLL expressions we define $\lambda =1/\theta$. This fixed relationship between $\theta$ and $\lambda$ will be used consistently in all subsequent formulas.

2.1. The Bivariate T-Y Family

The T-Y family of bivariate distributions provides a general and flexible framework for generating continuous bivariate models with given marginals and controlled dependence. This family is considered as an extension of the Farlie–Gumbel–Morgenstern (FGM) and Morgenstern-type constructions [17], where this approach is built upon two transformation functions T and Y that link the marginal and joint behaviors through an explicit dependence generator. The T-Y framework can be interpreted as a functional mechanism that allows one to preserve tractable analytical properties while extending the range of admissible dependence structures beyond linear or symmetric forms.

Building upon the foundational structure of the T-Y family, ref. [18] proposed an innovative method termed the (U,V)-YZ distribution family, designed for the construction of bivariate families of continuous distributions. To maintain consistency with the T-Y distributions in the marginal cases, they transformed the original T-Y approach, which was applicable to univariate or one-dimensional data, into a new generation mechanism suitable for bivariate or two-dimensional contexts. Let $r(u_1,u_2)$ and $R(u_1,u_2)$ represent, respectively, the probability density function PDF and the CDF of the random vector $(U_1,U_2)$, where $U_1\in [a_1,b_1]$, and $U_2\in [a_2,b_2]$, with $-\infty \le a_1<b_1\le \infty$, and $-\infty \le a_2<b_2\le \infty$. Now, assume that the upper limits ${\mathcal{D}}_1\left({\mathcal{G}}_1\left(y_1\right)\right)$ and ${\mathcal{D}}_2\left({\mathcal{G}}_2\left(y_2\right)\right)$ are two functions of the CDF ${\mathcal{G}}_1\left(y_1\right)$ and ${\mathcal{G}}_2\left(y_2\right)$ of random variables $Y_1$ and $Y_2$, respectively, that meet the following requirements:

$$\left\{\begin{array}{cc}\left(i\right)\hfill & D_1\left(G_1\left(y_1\right)\right)\in [a_1,b_1],D_2\left(G_2\left(y_2\right)\right)\in [a_2,b_2];\hfill \\ \left(ii\right)\hfill & D_1\left(G_1\left(y_1\right)\right)\mathrm{and}{D}_2\left(G_2\left(y_2\right)\right)\mathrm{are}\mathrm{differentiable}\mathrm{and}\mathrm{monotonically}\mathrm{non}-\mathrm{decreasing};\hfill \\ \left(iii\right)\hfill & D_1\left(G_1\left(y_1\right)\right)\to a_1\mathrm{as}{y}_1\to -\infty \mathrm{and}{D}_1\left(G_1\left(y_1\right)\right)\to b_1\mathrm{as}{y}_1\to \infty ;\hfill \\ \left(iv\right)\hfill & D_2\left(G_2\left(y_2\right)\right)\to a_2\mathrm{as}{y}_2\to -\infty \mathrm{and}{D}_2\left(G_2\left(y_2\right)\right)\to b_2\mathrm{as}{y}_2\to \infty .\hfill \end{array}\right.$$

Thus, according to [18], the CDF of a new bivariate family of distributions is defined as follows.

Definition 1. 

Let $Y_1$ (respectively, $Y_2$) be a random variable with $g_{Y_1}\left(y_1\right)$ (respectively, $g_{Y_2}\left(y_2\right)$) as PDF and ${\mathcal{G}}_{Y_1}\left(y_1\right)$ (respectively, ${\mathcal{G}}_{Y_2}\left(y_2\right)$) as CDF. Let $(U_1,U_2)$ be the continuous bivariate random vector with $r(u_1,u_2)$ as PDF and $\mathcal{C}(u_1,u_2)$ as CDF. Assume that $(U_1,U_2)$ is defined on $[a_1,b_1]\times [a_2,b_2]$. Hence, the CDF of the bivariate family is defined as

$$\mathcal{F}(y_1,y_2)={\int }_{-\infty }^{{\mathcal{D}}_1\left({\mathcal{G}}_1\left(y_1\right)\right)}{\int }_{-\infty }^{{\mathcal{D}}_2\left({\mathcal{G}}_2\left(y_2\right)\right)}r(u_1,u_2)d{u}_{1}d{u}_2,$$

(1)

which can be expressed as $\mathcal{F}(y_1,y_2)=C({\mathcal{D}}_1\left({\mathcal{G}}_1\left(y_1\right)\right),{\mathcal{D}}_2\left({\mathcal{G}}_2\left(y_2\right)\right))$.

Therefore, the corresponding PDF can be obtained by partially differentiating Equation (1) with respect to $y_1$ and $y_2$ as

$$f(y_1,y_2)={\displaystyle \frac{\partial {\mathcal{D}}_1\left({\mathcal{G}}_1\left(y_1\right)\right)}{\partial y_1}}{\displaystyle \frac{\partial {\mathcal{D}}_2\left({\mathcal{G}}_2\left(y_2\right)\right)}{\partial y_2}}\mathfrak{c}({\mathcal{D}}_1\left({\mathcal{G}}_1\left(y_1\right)\right),{\mathcal{D}}_2\left({\mathcal{G}}_2\left(y_2\right)\right)),$$

(2)

where $\mathfrak{c}(·,·)$ represents the copula density function.

2.2. Construction of a Bivariate Bounded Model Using Log-Logistic and Gompertz Distributions

Next, we introduce the bounded Gompertz–log-logistic (BGLL) distribution, which serves as the marginal component of the proposed BBGLL model. This construction combines two complementary properties:

• the Gompertz distribution’s capacity to model monotone exponential-like growth or decay; and
• the log-logistic distribution’s flexibility in generating unimodal or bathtub-shaped hazard rate functions.

By introducing an upper bound $\alpha$, the BGLL model captures realistic finite-support lifetimes that arise in practical systems with physical, biological, or contractual limits.

The Upper BGLL Distribution

The T-Y family (see Section 2.1) offers a functional approach for constructing bivariate distributions with flexible marginals and controlled dependence.

Let $Y_1$ (respectively, $Y_2$) be a random variable that follows the log-logistic distribution (LLD) with scale parameter $\alpha >0$ and a shape parameter equal to 1. Thus, the CDF of $Y_1$ and $Y_2$ are, respectively, given by

$${\mathcal{G}}_1(y_1\mid \alpha ,1)={\displaystyle \frac{y_1}{y_1+\alpha }}\mathrm{and}{\mathcal{G}}_2(y_2\mid \alpha ,1)={\displaystyle \frac{y_2}{y_2+\alpha }},$$

where $y_1>0$ and $y_2>0$. Now, by substituting the Gompertz distribution as the T-member and the log-logistic distribution as the Y-member into Equation (1), we obtain

$$\mathcal{F}\left(y_i\right)={\int }_0^{{\mathcal{D}}_i\left({\mathcal{G}}_i\left(y_i\right)\right)}{\displaystyle \frac{{\beta }^{1/\theta }}{{\theta }^2}}{\displaystyle \frac{e^{t/\theta }}{{\left(\beta -1+e^{t/\theta }\right)}^{1/\theta +1}}}dt,$$

(3)

where

$${\mathcal{D}}_i\left({\mathcal{G}}_i\left(y_i\right)\right)=log\left({\displaystyle \frac{1-{\mathcal{G}}_i\left(y_i\right)}{{\mathcal{G}}_i\left(y_i\right)}}\right)=log\left({\displaystyle \frac{\alpha }{y_i}}\right),i=1,2.$$

It should be noted that the form of the Gompertz model used in this construction corresponds to the generalized Gamma–Gompertz distribution commonly found in the reliability literature. Throughout the paper, we use the term “Gompertz distribution” for brevity, while the underlying functional form is the Gamma–Gompertz model. Hence, the survival function (SF) of the Gompertz distribution is given by

$$\mathbb{S}(t;\beta ,\theta )={\displaystyle \frac{{\beta }^{1/\theta }}{{(\beta -1+e^{t/\theta })}^{1/\theta }}},t>0,\beta >0,\theta >0.$$

(4)

Therefore, under a normalization constraint, Equation (3) yields a bounded CDF of the following form:

$$\mathcal{F}(y_i;\alpha ,{\beta }_i,\theta )={\displaystyle \frac{{\beta }_i^{1/\theta }}{{\left({\beta }_i-1+{\left(\frac{\alpha }{y_i}\right)}^{1/\theta }\right)}^{1/\theta }}},0<y_i<\alpha ,{\beta }_i>0,\theta >0,i=1,2.$$

(5)

Thus, one can see that

• As $y_i\downarrow 0$,

  $$\underset{y_i\to 0^{+}}{lim}\mathcal{F}\left(y_i\right)=\underset{y_i\to 0^{+}}{lim}{\displaystyle \frac{{\beta }^{1/\theta }}{{(\beta -1+{(\alpha /y_i)}^{1/\theta })}^{1/\theta }}}=0.$$
• As $y_i\uparrow \alpha$,

$$\underset{y_i\to {\alpha }^{-}}{lim}\mathcal{F}\left(y_i\right)={\displaystyle \frac{{\beta }^{1/\theta }}{{(\beta -1+1)}^{1/\theta }}}=1.$$

Hence, for $i=1,2$, $\mathcal{F}\left(y_i\right)$ correctly ranges from 0 to 1. Moreover, assume that $x_i={(\alpha /y_i)}^{1/\theta },i=1,2$. Then, we have

$${\displaystyle \frac{d{x}_i}{d{y}_i}}=-{\displaystyle \frac{1}{\theta }}{\displaystyle \frac{x_i}{y_i}}.$$

Differentiating $\mathcal{F}\left(y_i\right)={\beta }^{1/\theta }{(\beta -1+x_i)}^{-1/\theta }$, we obtain

$$f\left(y_i\right)={\mathcal{F}}^{\prime }\left(y_i\right)={\displaystyle \frac{{\beta }^{1/\theta }}{{\theta }^2}}{\displaystyle \frac{x_i}{y_i}}{(\beta -1+x_i)}^{-1/\theta -1},i=1,2.$$

Substituting $x_i={(\alpha /y_i)}^{1/\theta }$ gives

$$f\left(y_i\right)={\displaystyle \frac{{\beta }^{1/\theta }}{{\theta }^2}}{\displaystyle \frac{{\alpha }^{1/\theta }{y}_i^{-1/\theta -1}}{{(\beta -1+{(\alpha /y_i)}^{1/\theta })}^{1/\theta +1}}},0<y_i<\alpha ,i=1,2.$$

Since all components are non-negative for $\alpha >0,\beta >0,\theta >0$, it follows that $f\left(y_i\right)\ge 0$. Hence, $\mathcal{F}\left(y_i\right)$ is non-decreasing and right-continuous on $(0,\alpha )$, which confirms that the CDF presented in (5) is valid.

Differentiating Equation (5) with respect to $y_i$ and setting $\lambda =1/\theta$, we obtain

$$\begin{array}{cc}\hfill f_{Y_i}\left(y_i\right)& ={\displaystyle \frac{{\beta }_i^{\lambda }{\lambda }^2}{\alpha }}{\left({\displaystyle \frac{y_i}{\alpha }}\right)}^{-1-\lambda }{\left(-1+{\left({\displaystyle \frac{y_i}{\alpha }}\right)}^{-\lambda }+{\beta }_i\right)}^{-1-\lambda },0<y_i<\alpha ,\alpha ,{\beta }_i,\lambda >0,i=1,2.\hfill \end{array}$$

(6)

The above function is defined as the upper bounded Gompertz–log-logistic (BGLL) distribution with parameters $\alpha ,{\beta }_i$, and $\lambda$ that govern the distribution’s domain, scale, and shape, respectively.

Remark 1. 

The upper BGLL distribution is unimodal for certain parameter ranges and can exhibit heavy-tailed behavior depending on λ and β. Indeed, the logarithmic form of the PDF is

$$lnf\left(y_i\right)=2ln\lambda +\lambda ln\beta +\lambda ln\alpha -(\lambda +1)ln{y}_i-(\lambda +1)ln\left(\beta -1+{\left({\displaystyle \frac{\alpha }{y_i}}\right)}^{\lambda }\right).$$

(7)

Differentiating both sides of Equation (7) with respect to $y_i$ and setting it equal to zero yields the following equation:

$${\displaystyle \frac{d}{d{y}_i}}lnf\left(y_i\right)=-{\displaystyle \frac{\lambda +1}{y_i}}+{\displaystyle \frac{(\lambda +1)\lambda {(\alpha /y_i)}^{\lambda }}{y_i\left(\beta -1+{(\alpha /y_i)}^{\lambda }\right)}}=0.$$

which is of the form ${\left(y_i\right)}_{\mathrm{mode}}=\alpha {\left({\displaystyle \frac{\lambda -1}{\beta -1}}\right)}^{1/\lambda }$, so the mode of the BGLL distribution is given by

$${\left(y_i\right)}_{\mathrm{mode}}=\left\{\begin{array}{cc}\alpha {\left({\displaystyle \frac{\lambda -1}{\beta -1}}\right)}^{1/\lambda }\hfill & \mathrm{if}\lambda >1\mathrm{and}\beta >1,\hfill \\ \mathrm{Not}\mathrm{Exist}\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Note that, when the mode does not exist, the PDF is strictly monotonic:

• It decreases when $\lambda \le 1$ (maximum at $y_i=0$).
• It increases when $\beta \le 1$ (maximum at $y_i=\alpha$).

Figure 1 shows the behavior of the PDF with different sets of parameters, and points to the mentioned cases related to the mode of distribution.

Remark 2. 

The hazard rate function, HRF (also called failure rate function), related to the BGLL distribution is defined as $h_i={\displaystyle \frac{f\left(y_i\right)}{\mathcal{S}\left(y_i\right)}}$, where $\mathcal{S}\left(y_i\right)=1-\mathcal{F}\left(y_i\right)$ denotes the survival function. After simplification we get

$$h\left(y_i\right)={\displaystyle \frac{{\theta }^2{\beta }^{\lambda }{\alpha }^{\lambda }{y}_i^{-\lambda -1}}{{\left(\beta -1+{\left(\frac{\alpha }{y_i}\right)}^{\lambda }\right)}^{\lambda +1}-{\beta }^{\lambda }\left(\beta -1+{\left(\frac{\alpha }{y_i}\right)}^{\lambda }\right)}}.$$

Thus, one can see that

• as $y_i\to 0^{+}$: $h\left(y_i\right)\to 0(sincef\left(y_i\right)\to 0fasterthan\mathcal{S}\left(y_i\right)\to 1)$;
• as $y_i\to {\alpha }^{-}$: $h\left(y_i\right)\to {\displaystyle \frac{{\lambda }^2}{\alpha (\beta -1)}}\beta >1$.

Consequently, the HRF exhibits different behaviors depending on parameters:

• For $\lambda >1$ and $\beta >1$: HRF is initially increasing (IFR property) and may have a maximum point (unimodal).
• For $0<\lambda <1$ and $\beta <1$: HRF is bathtub (BTS property).
• For $\lambda =1$: h($y_i$) = ${\displaystyle \frac{1}{\alpha -y_i}}$ is strictly increasing.
• When $\beta =1$: $h\left(y_i\right)={\displaystyle \frac{\alpha }{y_i}}(decreasingforall\lambda >0)$ when $\lambda \to \infty$:

  $$h\left(y_i\right)\to \left\{\begin{array}{cc}0\hfill & fory<\alpha .\hfill \\ \infty \hfill & fory=\alpha .\hfill \end{array}\right.$$

The HRF described above is particularly useful for modeling systems with finite operational lifetimes, where the variable $y_i$ is less than or equal to a maximum lifespan α. It is also well-suited for components that exhibit various failure behaviors, including early-life failures characterized by a decreasing hazard rate, wear-out effects marked by an increasing hazard rate, or a combination of both, resulting in a bathtub-shaped hazard function; see Figure 2.

2.3. Bivariate Bounded Gompertz Log-Logistic Distribution

To capture dependence between two bounded lifetimes $Y_1$ and $Y_2$ (with the same support $(0,\alpha )$), we employ Sklar’s theorem, combining two BGLL marginals with the copula approach. Explicitly, by incorporating Equations (1), (2), (5), and (6) into Sklar’s theorem, the joint cumulative distribution function (CDF) of the BBGLL distribution for any chosen copula can be expressed as follows:

$$\begin{array}{c}\mathcal{F}(y_1,y_2)=\mathcal{C}\left({\displaystyle \frac{{\beta }_1^{\frac{1}{\theta }}}{{\left({\beta }_1-1+{\left(\frac{\alpha }{y_1}\right)}^{\frac{1}{\theta }}\right)}^{\frac{1}{\theta }}}},{\displaystyle \frac{{\beta }_2^{\frac{1}{\theta }}}{{\left({\beta }_2-1+{\left(\frac{\alpha }{y_2}\right)}^{\frac{1}{\theta }}\right)}^{\frac{1}{\theta }}}}\right),\hfill \\ {\beta }_1,{\beta }_2>0,\alpha ,\theta >0,0<y_1,y_2<\alpha .\hfill \end{array}$$

(8)

Thus, the related PDF can be expressed as follows:

$$\begin{array}{ccc}\hfill f(y_1,y_2)& =& {\displaystyle \frac{{\lambda }^4}{{\alpha }^2}}{\left({\displaystyle \frac{y_1{y}_2}{{\alpha }^2}}\right)}^{-(\lambda +1)}{\left({\beta }_1{\beta }_2\right)}^{\lambda }\prod _{i=1}^2{\left({\beta }_i-1+{\left({\displaystyle \frac{y_i}{\alpha }}\right)}^{-\lambda }\right)}^{-(\lambda +1)}\hfill \end{array}$$

$$\begin{array}{c}\times \mathfrak{c}\left({\displaystyle \frac{{\beta }_1^{\lambda }}{{\left({\beta }_1-1+{\left(\frac{\alpha }{y_1}\right)}^{\lambda }\right)}^{\lambda }}},{\displaystyle \frac{{\beta }_2^{\lambda }}{{\left({\beta }_2-1+{\left(\frac{\alpha }{y_2}\right)}^{\lambda }\right)}^{\lambda }}}\right),\hfill \\ {\beta }_1,{\beta }_2>0,\lambda (={\displaystyle \frac{1}{\theta }})>0,0<y_1,y_2<\alpha .\hfill \end{array}$$

(9)

3. Bivariate Bounded Gompertz–Log-Logistic Distribution with Clayton Copula

3.1. The BBGLL Construction with Clayton Copula

Let $Y_1$ and $Y_2$ be bounded lifetime variables both defined on $(0,\alpha )$, with marginal CDFs and PDFs that follow the BGLL form defined earlier. To introduce dependence, we apply Sklar’s theorem with the Clayton copula $C:{[0,1]}^2\to [0,1]$

$$C(u,v)={\left(u^{-\theta }+v^{-\theta }-1\right)}^{-1/\theta },\theta >0.$$

(10)

Although $\theta$ governs the dependence structure, in the present parametrization we set $\lambda =1/\theta$ for identifiability. Thus, $\theta$ is not independent of $\lambda$, and the marginal parameter $\lambda$ is entirely determined by $\theta$.

Remark 3. 

It is worth noting that the BBGLL construction is not restricted to the use of the Clayton copula. We focus on the Clayton family because it captures positive association and lower-tail dependence, which are natural features in reliability applications involving early-failure clustering. However, other Archimedean copulas could also be incorporated within the same framework. For instance, the Gumbel copula emphasizes upper-tail dependence, while the Frank copula yields symmetric dependence without tail concentration. Exploring these alternatives would provide BBGLL variants tailored to different dependence structures, and we identify this as a direction for future research.

Thus, based on (8), the CDF of the BBGLL distribution, with Clayton as the copula function, can be expressed as follows:

$${\mathcal{F}}_{Y_1,Y_2}(y_1,y_2)={\left({\displaystyle \frac{{\beta }_2\left({\beta }_1-1+{\left(\frac{\alpha }{y_1}\right)}^{1/\theta }\right)+{\beta }_1\left({\beta }_2-1+{\left(\frac{\alpha }{y_2}\right)}^{1/\theta }\right)-{\beta }_1{\beta }_2}{{\beta }_1{\beta }_2}}\right)}^{-{\displaystyle \frac{1}{\theta }}},$$

for $0<y_1,y_2<\alpha$. Now, replacing ${\displaystyle \frac{1}{\theta }}$ by $\lambda$, one can see that

$$\begin{array}{c}{\mathcal{F}}_{Y_1,Y_2}(y_1,y_2)={\left({\displaystyle \frac{{\beta }_2\left({\beta }_1-1+{\left(\frac{\alpha }{y_1}\right)}^{\lambda }\right)+{\beta }_1\left({\beta }_2-1+{\left(\frac{\alpha }{y_2}\right)}^{\lambda }\right)-{\beta }_1{\beta }_2}{{\beta }_1{\beta }_2}}\right)}^{-\lambda },\hfill \\ {\beta }_1,{\beta }_2>0,\lambda (={\displaystyle \frac{1}{\theta }})>0,0<y_1,y_2<\alpha .\hfill \end{array}$$

(11)

Theorem 1. 

The joint probability density function (PDF) of the BBGLL model is given by

$$f(y_1,y_2)={\displaystyle \frac{{\alpha }^2(1+\lambda ){\lambda }^2{\left(\frac{\alpha }{y_1}\right)}^{-1+\lambda }{\left(\frac{\alpha }{y_2}\right)}^{-1+\lambda }{\left(1-\frac{1-{\left(\frac{\alpha }{y_1}\right)}^{\lambda }}{{\beta }_1}-\frac{1-{\left(\frac{\alpha }{y_2}\right)}^{\lambda }}{{\beta }_2}\right)}^{-2-\lambda }}{{\beta }_1{\beta }_2{y}_1^2{y}_2^2}},$$

(12)

for ${\beta }_1,{\beta }_2>0,\lambda (={\displaystyle \frac{1}{\theta }})>0,0<y_1,y_2<\alpha .$

Proof. 

Based on Sklar’s theorem, the PDF of BBGLL model is given by

$$f_{Y_1,Y_2}(y_1,y_2)=\mathfrak{c}(u,v)·f_{Y_1\left(y_1\right)}·f_{Y_2\left(y_2\right)},$$

where $\mathfrak{c}(u,v)$ is the Clayton copula density defined by

$$\mathfrak{c}(u,v)={\displaystyle \frac{{\partial }^{2}C(u,v)}{\partial u\partial v}}=(1+\theta ){\left(uv\right)}^{-\theta -1}{(u^{-\theta }+v^{-\theta }-1)}^{-2-1/\theta }.$$

Thus, we obtain

$$f_{Y_1,Y_2}(y_1,y_2)={\displaystyle \frac{{\lambda }^2(\lambda +1){\alpha }^{2\lambda }·y_1^{-\lambda -1}{y}_2^{-\lambda -1}}{{\left({\beta }_1{\beta }_2\right)}^2}}·{\left({\displaystyle \frac{A(y_1,y_2)}{{\beta }_1{\beta }_2}}\right)}^{-\lambda -2},$$

where

$$A(y_1,y_2)={\beta }_2\left({\beta }_1-1+{\left({\displaystyle \frac{\alpha }{y_1}}\right)}^{\lambda }\right)+{\beta }_1\left({\beta }_2-1+{\left({\displaystyle \frac{\alpha }{y_2}}\right)}^{\lambda }\right)-{\beta }_1{\beta }_2.$$

Suitable simplifications of the above expression complete the proof. □

Remark 4. 

To analyze the boundary behavior of the joint PDF in (12), let $y_i=\alpha -{\epsilon }_i$ with ${\epsilon }_i\to 0^{+}$. A first-order expansion gives ${(\alpha /y_i)}^{\lambda }=1+O\left({\epsilon }_i\right)$, so the expression inside the brackets of Equation (12) satisfies

$$1-{\displaystyle \frac{1-{(\alpha /y_1)}^{\lambda }}{{\beta }_1}}-{\displaystyle \frac{1-{(\alpha /y_2)}^{\lambda }}{{\beta }_2}}=1+o({\epsilon }_1+{\epsilon }_2).$$

Therefore, the constant in the equivalent form

$${(constant+o({\epsilon }_1+{\epsilon }_2))}^{-\lambda -2}$$

is equal to 1. This shows that $f(y_1,y_2)$ approaches a finite, nonzero limit as $y_1,y_2\uparrow \alpha$.

Dependence structure governed by the $\beta$ and $\lambda$ parameters. The asymptotic properties of the given joint PDF can be analyzed by examining its behavior as the variables approach the boundaries of their support ($y_1,y_2\to 0^{+}$ and $y_1,y_2\to {\alpha }^{-}$) and as the parameters approach extreme values. As $y_1\to 0^{+}$ we get

$$f(y_1,y_2)\sim {\displaystyle \frac{{\lambda }^3(\lambda +1){\alpha }^{2\lambda }{\left(\frac{\alpha }{y_1}\right)}^{\lambda -1}{\left(\frac{\alpha }{y_2}\right)}^{\lambda -1}}{{\beta }_1{\beta }_2{y}_1^2{y}_2^2}}\Rightarrow f(y_1,y_2)\propto y_1^{-\lambda -1}{y}_2^{-\lambda -1}(\mathrm{power}-\mathrm{law}\mathrm{decay}).$$

The PDF exhibits heavy-tailed behavior as $y_i\to 0^{+}$ and the tail index is governed by $\lambda$, i.e., larger values of $\lambda$ imply faster decay (lighter tails) and smaller $\lambda$ suggest slower decay (heavier tails). This implies that finite moments require $\lambda >r$ for the r-th moment to exist.

As $1-{\displaystyle \frac{1-{\left({\displaystyle \frac{\alpha }{y_i}}\right)}^{\lambda }}{{\beta }_i}}\approx 1-{\displaystyle \frac{1-\left(1+\lambda {ϵ}_i\right)}{{\beta }_i}},\mathrm{where}{ϵ}_i={\displaystyle \frac{y_i}{\alpha }}-1\to 0^{-}$, one can see that

$${\left({\displaystyle \frac{\alpha }{y_i}}\right)}^{\lambda }\approx 1+\lambda {ϵ}_i+{\displaystyle \frac{\lambda (\lambda -1)}{2}}{ϵ}_i^2+\cdots .$$

Thus, the joint PDF will become $f(y_1,y_2)\propto {\left[\mathrm{constant}+\mathcal{O}({ϵ}_1+{ϵ}_2)\right]}^{-\lambda -2}⟶\mathrm{finite}\mathrm{limit}$. It means the PDF approaches a finite, nonzero value as $y_i\to {\alpha }^{+}$ and no singularity exists at the upper bound (unlike the lower bound).

Moreover, for $\lambda \to \infty$, the term ${\left({\displaystyle \frac{\alpha }{y_i}}\right)}^{\lambda }$ dominates, causing the joint PDF to factorize, i.e., $f(y_1,y_2)\approx f_{Y_1}\left(y_1\right)·f_{Y_2}\left(y_2\right)$. As $\lambda$ becomes large, $Y_1$ and $Y_2$ become independent. For $\lambda \to 0^{+}$ the joint PDF retains non-trivial dependence even as the tails and coefficient of tail dependence becomes

$$\eta =\underset{u\to 1}{lim}\mathbb{P}\left(Y_2>{\mathcal{F}}^{-1}\left(u\right)\mid Y_1>{\mathcal{F}}^{-1}\left(u\right)\right)\propto {\lambda }^{-1}.$$

As ${\beta }_i\to \infty$ the PDF reduces to a product of univariate generalized Pareto distributions, and the dependence structure vanishes. As ${\beta }_i\to 0$ the PDF becomes concentrated near $y_i=\alpha$ with a singularity at the upper bound. Figure 3 below shows the behavior of the PDF with different values of parameters.

3.1.1. Dependence Measures

The Clayton copula is an Archimedean copula with generator ${\phi }_{\theta }\left(t\right)=t^{-\theta }-1$, and belongs to the class of positive quadrant-dependent (PQD) copulas.

Proposition 1. 

For the BBGLL model, the following measures hold:

(1) 

Kendall’s τ and Spearman’s ρ:

$$\tau ={\displaystyle \frac{\theta }{\theta +2}},\rho ={\displaystyle \frac{\theta }{\theta +3}}.$$

(2) 

As $\theta \to 0^{+}$, $\tau ,\rho \to 0$ (independence); as $\theta \to \infty$, $\tau ,\rho \to 1$ (perfect positive dependence).

Proof. 

These relations are standard properties of the Clayton copula (see, e.g., [6,14]). □

3.1.2. Tail Dependence and Tail Orders

Tail dependence quantifies the probability of simultaneous extremes. For the BBGLL model, lower tails correspond to early failures ($Y_i\to 0^{+}$), and upper tails correspond to lifetimes approaching their finite bound $\alpha$.

Let $(Y_1,Y_2)$ be a bivariate random vector with marginal CDFs ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$. The upper-tail-dependence coefficient ${\gamma }_U$ and lower-tail-dependence coefficient ${\gamma }_L$ are, respectively, defined by

$${\gamma }_U=\underset{q\to 1^{-}}{lim}\mathbb{P}(Y_2>{\mathcal{F}}_2^{-1}\left(q\right)\mid Y_1>{\mathcal{F}}_1^{-1}\left(q\right))\mathrm{and}{\gamma }_L=\underset{q\to 0^{+}}{lim}\mathbb{P}(Y_2\le {\mathcal{F}}_2^{-1}\left(q\right)\mid Y_1\le {\mathcal{F}}_1^{-1}\left(q\right)).$$

On the other hand, the upper-tail order ${\kappa }_U$ and lower-tail order ${\kappa }_L$ are, respectively, given by

$${\kappa }_U=\underset{q\to 1^{-}}{lim}{\displaystyle \frac{log\mathbb{P}(Y_1>{\mathcal{F}}_1^{-1}\left(q\right),Y_2>{\mathcal{F}}_2^{-1}\left(q\right))}{log(1-q)}}\mathrm{and}{\kappa }_L=\underset{q\to 0^{+}}{lim}{\displaystyle \frac{logP(Y_1\le F_1^{-1}\left(q\right),Y_2\le F_2^{-1}\left(q\right))}{logq}}.$$

Theorem 2. 

The PDF of BBGLL (with Clayton copula), defined by (12), has the following:

• Upper-tail dependence, i.e., ${\gamma }_U=0$ (asymptotic independence).
• Lower-tail dependence, i.e., ${\gamma }_L=0$.
• Upper-tail order, i.e., ${\kappa }_U=1+\lambda$ (intermediate tail dependence).
• Lower-tail order, i.e., ${\kappa }_L=\infty$ (strictly faster than any power-law decay).

Proof. 

For ${\gamma }_U$ and ${\kappa }_U$: first we have to transform $u_i={(\alpha /y_i)}^{\lambda }$ for $i=1,2$. We observe that near the upper tail ($y_i\to {\alpha }^{-}$), the joint survival satisfies the following: (i) $\mathbb{P}(Y_1>y,Y_2>y)\sim {(2u-2)}^{-1-\lambda }=2^{-1-\lambda }{(u-1)}^{-1-\lambda }$; (ii) the marginal survival $\mathbb{P}(Y_1>y)\sim {(u-1)}^{-\lambda }$. Thus, we have ${\gamma }_U={lim}_{u\to 1^{+}}{\displaystyle \frac{2^{-1-\lambda }{(u-1)}^{-1-\lambda }}{{(u-1)}^{-\lambda }}}=0$ and ${\kappa }_U=1+\lambda \mathrm{because}{\displaystyle \frac{logP(Y_1>y,Y_2>y)}{logP(Y_1>y)}}\to 1+\lambda$. Similarly, for ${\gamma }_L$ and ${\kappa }_L$, as $y_1,y_2\to 0^{+}$, $f(y_1,y_2)\to 0$ due to $0<y_1,y_2<\alpha$. No weight accumulates near $(0,0)$, giving ${\gamma }_L=0$ and ${\kappa }_L=\infty$. □

3.1.3. Positive Quadrant Dependence

Proposition 2. 

For any $\theta >0$, the BBGLL distribution is positively quadrant dependent (PQD), i.e.,

$$P(Y_1\le y_1,Y_2\le y_2)\ge P(Y_1\le y_1)P(Y_2\le y_2),\forall (y_1,y_2)\in (0,\alpha )\times (0,\alpha ).$$

Independence is achieved only when $\theta =0$.

Proof. 

The joint CDF, defined by (11), can be simplified to ${\left({\displaystyle \frac{{\beta }_1{\beta }_2}{N}}\right)}^{\lambda }$, where $N={\beta }_2{A}_1+{\beta }_1{A}_2-{\beta }_1{\beta }_2$, with $A_i={\beta }_i-1+{\left({\displaystyle \frac{\alpha }{y_i}}\right)}^{\lambda }$. Then, the marginal CDFs, defined by (5), can be expressed as ${\mathcal{F}}_{Y_i}\left(y_i\right)={\displaystyle \frac{{\beta }_i^{\lambda }}{A_i^{\lambda }}}$. To establish positive dependence, it suffices to verify that ${\mathcal{F}}_{Y_1,Y_2}(y_1,y_2)\ge {\mathcal{F}}_{Y_1}\left(y_1\right)\times {\mathcal{F}}_{Y_2}\left(y_2\right)$, which reduces to demonstrating that $A_1{A}_2\le N$.

By explicitly calculating $A_1{A}_2$ and N, it can be shown that their difference simplifies to $-(1-u)(1-v)$, where $u={\left({\displaystyle \frac{\alpha }{y_1}}\right)}^{\lambda }$ and $v={\left({\displaystyle \frac{\alpha }{y_2}}\right)}^{\lambda }$. Noting that for $y_1,y_2\ge \alpha$, both $u,v\le 1$, it follows that $(1-u),(1-v)\ge 0$, and thus their product is non-negative. Consequently, $N-A_1{A}_2\le 0$, implying $A_1{A}_2\le N$. This leads to the inequality ${\displaystyle \frac{1}{N}}\ge {\displaystyle \frac{1}{A_1{A}_2}}$, which in turn establishes that ${\mathcal{F}}_{Y_1,Y_2}(y_1,y_2)\ge {\mathcal{F}}_{Y_1}\left(y_1\right)\times {\mathcal{F}}_{Y_2}\left(y_2\right)$. Therefore, the joint distribution exhibits positive quadrant dependence in the region where $y_1,y_2\le \alpha$. □

3.1.4. Survival Function

To derive the survival function $\mathcal{S}(y_1,y_2)$, we consider the probability that both $Y_1>y_1$ and $Y_2>y_2$. For the survival copula $C_{\mathcal{S}}(u_1,u_2)$ to be valid in the Clayton, the condition is is expressed as ${(1-u_1)}^{-\theta }+{(1-u_2)}^{-\theta }-1\ge 0,$ and must hold for all $u_1,u_2\in [0,1]$. This is automatically satisfied for Clayton with $\theta >0$ for $u_1,u_2\in [0,1]$.

But here, $u_i={\mathcal{S}}_i\left(y_i\right)$ depends on $y_i<\alpha$, so ${\mathcal{S}}_i\left(y_i\right)>0$. Also, $u_i<0$ ensures $1-u_i>1$, so $u_i>0$. No further restriction links $y_1$ and $y_2$ except the natural one from the copula’s domain: in terms of $u_1,u_2$, ${\mathcal{C}}_{\mathcal{S}}={(1-u_1-u_2)}^{-\lambda }$ requires $1-u_1-u_2>0\Rightarrow u_1+u_2<1$. That is,

$${\beta }_1\left[1-{\left({\displaystyle \frac{\alpha }{y_1}}\right)}^{\lambda }\right]+{\beta }_2\left[1-{\left({\displaystyle \frac{\alpha }{y_2}}\right)}^{\lambda }\right]<1$$

Since the numerator $1-{\left({\displaystyle \frac{\alpha }{y_i}}\right)}^{\lambda }<0$, this inequality is not automatically true, it imposes a relation between $y_1$ and $y_2$ for the formula to be valid.

So, the relation is

$${\beta }_1\left[1-{\left({\displaystyle \frac{\alpha }{y_1}}\right)}^{\lambda }\right]+{\beta }_2\left[1-{\left({\displaystyle \frac{\alpha }{y_2}}\right)}^{\lambda }\right]<1$$

Now, the bivariate survival function is obtained by integrating the joint PDF over the region $(y_1,\infty )\times (y_2,\infty )$. Since the support is $0<y_1,y_2<\alpha$, the upper limits are $\alpha$:

$$\mathcal{S}(y_1,y_2)=\mathcal{P}(Y_1>y_1,Y_2>y_2)={\int }_{u=y_1}^{\alpha }{\int }_{v=y_2}^{\alpha }f(u,v)dvdu.$$

Given the joint CDF ${\mathcal{F}}_{Y_1,Y_2}(y_1,y_2)$, the survival function can also be expressed as

$$\mathcal{S}(y_1,y_2)=1-{\mathcal{F}}_{Y_1}\left(y_1\right)-{\mathcal{F}}_{Y_2}\left(y_2\right)+{\mathcal{F}}_{Y_1,Y_2}(y_1,y_2).$$

Thus, the survival function simplifies to

$$\begin{array}{ccc}\hfill \mathcal{S}(y_1,y_2)& =& 1-{\left(1-{\displaystyle \frac{1-{\left(\frac{\alpha }{y_1}\right)}^{\lambda }}{{\beta }_1}}\right)}^{-\lambda }-{\left(1-{\displaystyle \frac{1-{\left(\frac{\alpha }{y_2}\right)}^{\lambda }}{{\beta }_2}}\right)}^{-\lambda }\hfill \\ & +& {\left(1-{\displaystyle \frac{1-{\left(\frac{\alpha }{y_1}\right)}^{\lambda }}{{\beta }_1}}-{\displaystyle \frac{1-{\left(\frac{\alpha }{y_2}\right)}^{\lambda }}{{\beta }_2}}\right)}^{-\lambda },\hfill \end{array}$$

where ${\beta }_1,{\beta }_2>0,\lambda (={\displaystyle \frac{1}{\theta }})>0,0<y_1,y_2<\alpha .$

To verify the properties of the survival function $\mathcal{S}(y_1,y_2)$, we begin by checking the bounds. As $y_1\to 0^{+}$ and $y_2\to 0^{+}$, we observe that ${\left({\displaystyle \frac{\alpha }{y_i}}\right)}^{\lambda }\to \infty$, implying each transformed term in the function tends to zero. Hence, the survival function simplifies to $\mathcal{S}(0,0)=1-0-0+0=1$, satisfying the upper-bound condition. Next, as $y_1\to {\alpha }^{-}$ and $y_2\to {\alpha }^{-}$, it follows that ${\left({\displaystyle \frac{\alpha }{y_i}}\right)}^{\lambda }\to 1$. Substituting into the survival function, the expression reduces to $\mathcal{S}(y_1,y_2)=1-{(1-0)}^{-\lambda }-{(1-0)}^{-\lambda }+{(1-0-0)}^{-\lambda }=1-1-1+1=0$, confirming the lower bound.

To verify monotonicity, we note that the terms of the form ${\left(1-{\displaystyle \frac{1-{\left(\frac{\alpha }{y_i}\right)}^{\lambda }}{{\beta }_i}}\right)}^{-\lambda }$ are non-decreasing in $y_i$, because ${\left({\displaystyle \frac{\alpha }{y_i}}\right)}^{\lambda }$ is a decreasing function of $y_i$. Therefore, $S(y_1,y_2)$ is non-increasing in both $y_1$ and $y_2$, which is consistent with the behavior expected of a survival function.

Regarding the marginal survival functions, substituting $y_2=0$ directly into $\mathcal{S}(y_1,y_2)$ is incorrect due to improper handling of boundary behavior. Instead, we use the following limiting process:

$${\mathcal{S}}_{Y_1}\left(y_1\right)=\underset{y_2\to 0^{+}}{lim}\mathcal{S}(y_1,y_2)=1-{\left(1-{\displaystyle \frac{1-{\left(\frac{\alpha }{y_1}\right)}^{\lambda }}{{\beta }_1}}\right)}^{-\lambda },$$

which provides a valid univariate survival function for $Y_1$. Similarly, the marginal survival function for $Y_2$ is

$${\mathcal{S}}_{Y_2}\left(y_2\right)=1-{\left(1-{\displaystyle \frac{1-{\left(\frac{\alpha }{y_2}\right)}^{\lambda }}{{\beta }_2}}\right)}^{-\lambda }.$$

Thus, both marginal functions are mathematically valid and consistent with survival function properties. Figure 4 shows the behavior of the bivariate SF with different sets of parameters.

3.1.5. Hazard Rate Function

The joint hazard rate associated with the BBGLL distribution is defined as

$$h(y_1,y_2)={\displaystyle \frac{f(y_1,y_2)}{S(y_1,y_2)}},(y_1,y_2)\in {(0,\alpha )}^2,$$

where $f(y_1,y_2)$ is the joint density and $S(y_1,y_2)=\mathbb{P}(Y_1>y_1,Y_2>y_2)$ is the joint survival function. Because the model is supported on the bounded domain ${(0,\alpha )}^2$, the hazard exhibits characteristic boundary behaviors that we detail below.

Behavior of the joint hazard rate as $y_i\to 0^{+}$. From the expression of the joint density in Theorem 1, we obtain the following approximation:

$$f(y_1,y_2)\sim C_0{y}_1^{-(1+\lambda )}{y}_2^{-(1+\lambda )},y_i\to 0^{+},$$

for a positive constant $C_0$ depending on $(\alpha ,{\beta }_1,{\beta }_2,\lambda )$. Since the joint survival satisfies

$$S(y_1,y_2)\to 1,y_i\to 0^{+},$$

the joint hazard diverges:

$$h(y_1,y_2)\to +\infty ,y_i\to 0^{+}.$$

This divergence occurs only in an extremely narrow neighborhood of the origin. For the values of the parameters used in Figure 5, numerical evaluation shows that $h(y_1,y_2)$ becomes extremely large only when $y_i<0.01\alpha$. Such a region is too narrow to be represented without flattening the remaining portion of the surface.

Behavior of the joint hazard rate as $y_i\to {\alpha }^{-}$. As $y_i$ approaches the upper bound $\alpha$, we have

$$f(y_1,y_2)\to C_{\alpha }>0,y_i\to {\alpha }^{-},$$

for some finite constant $C_{\alpha }$, while the survival function vanishes:

$$S(y_1,y_2)\to 0,y_i\to {\alpha }^{-}.$$

Therefore,

$$h(y_1,y_2)\to +\infty ,y_i\to {\alpha }^{-}.$$

Again, the divergence occurs only in a very thin region near the boundary. Within the interior domain, the hazard remains finite and evolves smoothly.

Remark 5. 

Figure 5 illustrates the shape of the joint hazard over the practically relevant domain. For readability, the hazard is plotted only on the truncated region ${[0.5,\alpha -0.01]}^2$ rather than on the full interval ${(0,\alpha )}^2$. This avoids the numerical blow-up occurring extremely close to the boundaries and highlights the behavior over the interior of the domain. The apparent absence of divergence in the figure is therefore consistent with the asymptotic limits derived above.

Remark 6. 

The divergence of the joint hazard near the boundaries is a structural consequence of the bounded support of the BBGLL model. However, these extreme effects occur only in vanishingly small neighborhoods of 0 and α. Outside these thin regions, the hazard function remains smooth, as clearly visible in Figure 5. This explains why the asymptotic blow-up, although theoretically correct, is not visible within the plotted range.

3.1.6. Conditional Densities

The conditional densities for $Y_2|y_1$ and $Y_1|y_2$ are obtained, respectively, as

$$f\left(y_2\right|y_1)={\displaystyle \frac{f(y_1,y_2)}{f_{Y_1}\left(y_2\right)}}\mathrm{and}f\left(y_1\right|y_2)={\displaystyle \frac{f(y_1,y_2)}{f_{Y_2}\left(y_2\right)}},$$

Now, based on (6) and (12), we obtain

$$\begin{array}{c}f\left(y_2\right|y_1)={\displaystyle \frac{(1+\lambda )\lambda {\alpha }^{\lambda }{\left(\frac{\alpha }{y_2}\right)}^{\lambda -1}{\left(1-\frac{1-{(\alpha /y_1)}^{\lambda }}{{\beta }_1}-\frac{1-{(\alpha /y_2)}^{\lambda }}{{\beta }_2}\right)}^{-2-\lambda }}{y_2^2{\beta }_2{\left(1-\frac{1-{(\alpha /y_1)}^{\lambda }}{{\beta }_1}\right)}^{-1-\lambda }}}\\ \hfill {\beta }_1,{\beta }_2>0,\lambda (={\displaystyle \frac{1}{\theta }})>0,0<y_1,y_2<\alpha .\end{array}$$

and

$$\begin{array}{c}f\left(y_1\right|y_2)={\displaystyle \frac{(1+\lambda )\lambda {\alpha }^{\lambda }{\left(\frac{\alpha }{y_1}\right)}^{\lambda -1}{\left(1-\frac{1-{(\alpha /y_1)}^{\lambda }}{{\beta }_1}-\frac{1-{(\alpha /y_2)}^{\lambda }}{{\beta }_2}\right)}^{-2-\lambda }}{y_1^2{\beta }_1{\left(1-\frac{1-u_2}{{\beta }_2}\right)}^{-1-\lambda }}}\\ \hfill {\beta }_1,{\beta }_2>0,\lambda (={\displaystyle \frac{1}{\theta }})>0,0<y_1,y_2<\alpha .\end{array}$$

The conditional distributions exhibit power-law behavior, modified by a dependence term inversely proportional to the square of one minus the sum of the reciprocals of the parameters ${\beta }_1$ and ${\beta }_2$ and a power of $\lambda$. This dependence term asymmetrically influences the distributions, with ${\beta }_1$ and ${\beta }_2$ having differing effects. As the exponent $\lambda$ approaches zero from the positive side, the conditional distributions approach independence.

Remark 7. 

Based on the conditional method, one can generate a bivariate sample from the BBGLL distribution by first simulating a pair $(U_1,U_2)$ from the Clayton copula and then applying the inverse BGLL marginal CDFs.

For the Clayton copula

$$C_{\theta }(u_1,u_2)={\left(u_1^{-\theta }+u_2^{-\theta }-1\right)}^{-1/\theta },$$

the conditional distribution of $U_2$ given $U_1=u_1$ is obtained from

$$C(u_2\mid u_1)={\displaystyle \frac{\partial }{\partial u_1}}{C}_{\theta }(u_1,u_2)=u_1^{-(1+\theta )}{C}_{\theta }{(u_1,u_2)}^{1+\theta }.$$

Setting $C(u_2\mid u_1)=W$ with $W\sim \mathrm{Uniform}(0,1)$ gives the identity

$$C_{\theta }(u_1,u_2)=u_1{W}^{1/(1+\theta )}.$$

Solving this equation for $u_2$ yields the explicit inversion formula

$$u_2={\left[1+\left(u_1^{-\theta }-1\right)\left(W^{-\theta /(1+\theta )}-1\right)\right]}^{-1/\theta },$$

which justifies the expression used in the second step of the algorithm below. This shows that solving $C(u_2\mid u_1)=W$ naturally leads to the identity $C_{\theta }(u_1,u_2)=u_1{W}^{1/(1+\theta )}$ and explains the origin of the formula for generating $U_2$.

The simulation steps are therefore as follows:

• Generate $U_1,W\sim Uniform(0,1)$.
• Compute

  $$U_2={\left[1+\left(U_1^{-\theta }-1\right)\left(W^{-\theta /(1+\theta )}-1\right)\right]}^{-1/\theta }.$$
• Compute the BGLL marginal values

  $$y_1=\alpha {\left[{\beta }_1({(1-U_1)}^{-1/\lambda }-1)+1\right]}^{-1/\lambda },y_2=\alpha {\left[{\beta }_2({(1-U_2)}^{-1/\lambda }-1)+1\right]}^{-1/\lambda }.$$
• Return $(y_1,y_2)$.

3.2. Main Properties

Moment Properties

In this section, we focus on the tractable analytical properties of the BBGLL model. In particular, the moment-based characterizations lead to interpretable structural insights. Moments are key indicators of the location, dispersion, and dependence characteristics of a probability distribution.

Let $(Y_1,Y_2)$ follow the BBGLL distribution introduced in Section 2. To study its moment structure, we consider the joint $(r,s)$th moment

$$E\left(Y_1^r{Y}_2^s\right)={\int }_0^{\alpha }{\int }_0^{\alpha }{y}_1^r{y}_2^{s}f(y_1,y_2)d{y}_{1}d{y}_2,$$

which is finite for all non-negative integers $r,s$ due to the bounded support ${(0,\alpha )}^2$. Using the expression of the joint density in Theorem 1, and applying the transformation $v_i={(\alpha /y_i)}^{\lambda }$, the above integral can be rewritten as

$$E\left(Y_1^r{Y}_2^s\right)={\alpha }^{r+s}{(1+\lambda )}^{\lambda }{\beta }_1^{-r}{\beta }_2^{-s}{c}^{-(\lambda +r+s)}J\left({\displaystyle \frac{1}{{\beta }_{1}c}},{\displaystyle \frac{1}{{\beta }_{2}c}};1+{\displaystyle \frac{r}{\lambda }},1+{\displaystyle \frac{s}{\lambda }},\lambda +r+s\right),$$

where

$$c=1-{\displaystyle \frac{1}{{\beta }_1}}-{\displaystyle \frac{1}{{\beta }_2}},$$

and the bivariate integral $J(a,b;p,q,\gamma )$ is defined by

$$J(a,b;p,q,\gamma )={\int }_a^{\infty }{\int }_b^{\infty }{v}_1^{-p}{v}_2^{-q}{(1+v_1+v_2)}^{-\gamma }d{v}_{1}d{v}_2.$$

This integral admits a closed representation in terms of the Appell hypergeometric function $F_1$:

$$\begin{array}{c}\hfill J(a,b;p,q,\gamma )=a^{1-p}{b}^{1-q}{(a+b+1)}^{p+q-\gamma -2}B(1-p,\gamma -p-q+1)\\ \hfill \times F_1\left(1-p;\gamma -p-q+1,1-q;2-\gamma ;-{\displaystyle \frac{1}{a}},-{\displaystyle \frac{1}{b}}\right),\end{array}$$

where $B(·,·)$ denotes the Beta function. Substituting this expression into the transformed moment integral yields the general formula for $E\left(Y_1^r{Y}_2^s\right)$ for the BBGLL model.

Sanity check: the case $r=s=0$.

Setting $r=s=0$ in the previous expression gives

$$E\left(1\right)={(1+\lambda )}^{\lambda }{c}^{-\lambda }J\left(\frac{1}{{\beta }_{1}c},\frac{1}{{\beta }_{2}c};1,1,\lambda \right)=1,$$

confirming that the joint density integrates to unity.

Remark 8. 

The appearance of the Appell hypergeometric function $F_1$ arises naturally from the structure of the BBGLL joint density, which combines bounded log-logistic components with a Gamma–Gompertz transformation. Although the closed-form expressions provide compact formulas for the moments, their numerical evaluation typically requires computer algebra systems such as Mathematica 13.0 and Python 3.12.4 with Jupyter Notebook. These tools offer stable implementations of $F_1$ for all parameter ranges considered in this work, ensuring that the moment expressions are fully tractable in practice.

3.3. Estimation Methods

3.3.1. Maximum Likelihood Estimation (MLE)

The maximum likelihood estimation (MLE) is a fundamental method in statistical inference used to estimate the parameters of a probability distribution by maximizing the likelihood function based on observed data. For a set of n independent and identically distributed observations $(y_{1i},y_{2i})$, the likelihood function $L(\alpha ,{\beta }_1,{\beta }_2,\lambda )$ represents the joint probability of observing the data given the parameters. It is expressed as the product of the joint probability density functions (PDFs) evaluated at each data point:

$$L(\alpha ,{\beta }_1,{\beta }_2,\lambda )=\prod _{i=1}^{n}f(y_{1i},y_{2i}\mid \alpha ,{\beta }_1,{\beta }_2,\lambda ).$$

Substituting the explicit form of the joint PDF yields a complex product involving the parameters and data points. To facilitate differentiation and optimization, it is common to work with the log-likelihood function, which converts the product into a sum:

$$\ell (\alpha ,{\beta }_1,{\beta }_2,\lambda )=nln\left({\displaystyle \frac{(1+\lambda ){\lambda }^3}{{\beta }_1{\beta }_2}}\right)+2\lambda nln\alpha -(1+\lambda )\sum _{i=1}^n(ln{y}_{1i}+ln{y}_{2i})-(2+\lambda )\sum _{i=1}^{n}ln{u}_i,$$

(13)

where

$$u_i=1-{\displaystyle \frac{1-{\left(\frac{\alpha }{y_{1i}}\right)}^{\lambda }}{{\beta }_1}}-{\displaystyle \frac{1-{\left(\frac{\alpha }{y_{2i}}\right)}^{\lambda }}{{\beta }_2}}.$$

Maximizing the log-likelihood involves taking its derivatives with respect to each parameter and setting them to zero, leading to a system of nonlinear equations. For example, the partial derivative with respect to $\alpha$ accounts for the sensitivity of the likelihood to changes in this scale parameter, incorporating terms involving the data and the parameters. Similarly, derivatives with respect to ${\beta }_1$, ${\beta }_2$, and $\lambda$ involve intricate expressions that include logarithmic and power functions of the data.

Given the nonlinearity and complexity of these equations, analytical solutions are generally infeasible. Instead, numerical optimization algorithms—such as the Newton–Raphson or quasi-Newton methods (e.g., BFGS)—are employed to find the parameter estimates that maximize the likelihood. These methods iteratively update parameter values based on gradient and Hessian information, converging to the maximum likelihood estimates under suitable conditions. Constraints such as $\alpha >max(y_{1i},y_{2i})$ and positivity of ${\beta }_1,{\beta }_2,\lambda$ are typically enforced to ensure meaningful solutions. On the other hand, we have used the process presented in Remark 7 for the generation of bivariate random numbers. In practice, statistical software like Python 3.12.4 with Jupyter Notebook provides robust implementations of these optimization routines, facilitating efficient and accurate estimation of the model parameters. We have computed their average bias ($\mathbb{E}(\widehat{\Theta }-\Theta )$), mean squared error (MSE) ($\mathbb{E}{(\widehat{\Theta }-\Theta )}^2$), 95% confidence intervals ($\mathbb{E}\left(\widehat{\Theta }\right)\pm 1.96\times \sqrt{\mathrm{Var}\left(\widehat{\Theta }\right))}$), and probability of convergence for the maximum likelihood estimators (MLEs). Since the marginal distributions are not standard, we conducted the simulation by adopting the following strategy:

• Simulate pairs $(u,v)$ from a Clayton copula using the conditional distribution method.
• Transform $u,v$ to $y_1,y_2$ using the inverse marginal CDFs.
• Estimate MLEs for each sample.
• Compute averages, biases, and MSEs across replicates.
• Generate plots for bias and MSE.

The inverse marginal CDF for $y_1$ is derived from

$$\mathcal{F}\left(y_1\right)=u\Rightarrow u={\beta }_1^{\lambda }{\left({\beta }_1-1+{\alpha }^{\lambda }{y}_1^{-\lambda }\right)}^{-\lambda },$$

which implies ${\left({\beta }_1-1+{\alpha }^{\lambda }{y}_1^{-\lambda }\right)}^{\lambda }\Rightarrow {\displaystyle \frac{{\beta }_1^{\lambda }}{u}}$ and ${\alpha }^{\lambda }{y}_1^{-\lambda }={\beta }_1{u}^{-1/\lambda }-{\beta }_1+1\Rightarrow y_1^{-\lambda }={\displaystyle \frac{{\beta }_1{u}^{-1/\lambda }-{\beta }_1+1}{{\alpha }^{\lambda }}}$; thus $y_1=\alpha {\left({\beta }_1{u}^{-1/\lambda }-{\beta }_1+1\right)}^{-1/\lambda }$. Similarly for $y_2$ with parameter ${\beta }_2$, we have $y_2=\alpha {\left({\beta }_2{v}^{-1/\lambda }-{\beta }_2+1\right)}^{-1/\lambda }$.

3.3.2. Inference Functions for Margins (IFM) Estimation

The IFM method involves two stages: estimating the marginal parameters and then the copula parameter (if applicable). Since $\lambda$ is shared between the margins and the copula, we primarily estimate it via the margins.

For a sample $\{y_{i1},y_{i2},\dots ,y_{in}\}$ for $Y_i$ ($i=1,2$), the log-likelihood for the marginal distribution is

$${\ell }_i(\alpha ,{\beta }_i,\lambda )=\sum _{j=1}^{n}log{f}_{Y_i}(y_{ij};\alpha ,{\beta }_i,\lambda ).$$

The log-PDF is

$$log{f}_{Y_i}\left(y_{ij}\right)=\lambda log{\beta }_i+2log\lambda +\lambda log\alpha -(1+\lambda )log{y}_{ij}-(1+\lambda )log\left({\beta }_i-1+{\left({\displaystyle \frac{\alpha }{y_{ij}}}\right)}^{\lambda }\right).$$

The combined marginal log-likelihood is

$$\ell (\alpha ,{\beta }_1,{\beta }_2,\lambda )={\ell }_1(\alpha ,{\beta }_1,\lambda )+{\ell }_2(\alpha ,{\beta }_2,\lambda ).$$

The IFM estimates $(\widehat{\alpha },{\widehat{\beta }}_1,{\widehat{\beta }}_2,\widehat{\lambda })$ are obtained by maximizing

$$(\widehat{\alpha },{\widehat{\beta }}_1,{\widehat{\beta }}_2,\widehat{\lambda })=arg\underset{\alpha ,{\beta }_1,{\beta }_2,\lambda }{max}\ell (\alpha ,{\beta }_1,{\beta }_2,\lambda ),$$

subject to $\alpha >max(y_{1j},y_{2j})$, ${\beta }_1,{\beta }_2>0$, and $\lambda >0$. Since $\lambda$ is estimated in the marginal step, we may use $\widehat{\lambda }$ directly in the Clayton copula. Optionally, to refine $\lambda$, we may compute pseudo-observations:

$$u_{ij}=F(y_{ij};\widehat{\alpha },{\widehat{\beta }}_i,\widehat{\lambda }),$$

and maximize the copula log-likelihood:

$${\ell }_c\left(\lambda \right)=\sum _{j=1}^n\left[log(1+\lambda )-(1+\lambda )log\left(u_{1j}{u}_{2j}\right)-(2+\lambda )log\left(u_{1j}^{-\lambda }+u_{2j}^{-\lambda }-1\right)\right].$$

The IFM estimates are obtained by numerically maximizing the combined marginal log-likelihood, subject to the specified constraints. If needed, the copula log-likelihood can be maximized to refine $\lambda$. Numerical optimization (e.g., Newton–Raphson or gradient-based methods) is recommended due to the complexity of the expressions.

3.3.3. Semi-Parametric Estimation Procedure

The semi-parametric (SP) method estimates the marginal distributions non-parametrically and the copula parameter parametrically, with adjustments for shared parameters. For a sample $\{y_{i1},y_{i2},\dots ,y_{in}\}$ for $Y_i$ ($i=1,2$), the empirical CDF is

$${\widehat{F}}_i\left(y\right)={\displaystyle \frac{1}{n}}\sum _{j=1}^n\mathbb{I}(y_{ij}\le y).$$

Pseudo-observations are computed as

$$u_{ij}={\widehat{F}}_i\left(y_{ij}\right)={\displaystyle \frac{\mathrm{rank}\left(y_{ij}\right)}{n+1}},$$

using rank-based scaling to avoid boundary issues. The Clayton copula parameter $\lambda$ is estimated by maximizing the copula log-likelihood (see [19]):

$${\ell }_c\left(\lambda \right)=\sum _{j=1}^n\left[log(1+\lambda )-(1+\lambda )log\left(u_{1j}{u}_{2j}\right)-(2+\lambda )log\left(u_{1j}^{-\lambda }+u_{2j}^{-\lambda }-1\right)\right],$$

subject to $\lambda >0$, using numerical optimization. Given the estimated $\widehat{\lambda }$, the marginal parameters $\alpha$ and ${\beta }_i$ are estimated by maximizing the marginal log-likelihood for each $Y_i$:

$${\ell }_i(\alpha ,{\beta }_i;\widehat{\lambda })=\sum _{j=1}^n\left[\widehat{\lambda }log{\beta }_i+2log\widehat{\lambda }+\widehat{\lambda }log\alpha -(1+\widehat{\lambda })log{y}_{ij}-(1+\widehat{\lambda })log\left({\beta }_i-1+{\left({\displaystyle \frac{\alpha }{y_{ij}}}\right)}^{\widehat{\lambda }}\right)\right].$$

Since $\alpha$ is shared, estimate ${\alpha }_1$ and ${\alpha }_2$ for each margin and compute the average:

$$\widehat{\alpha }={\displaystyle \frac{{\widehat{\alpha }}_1+{\widehat{\alpha }}_2}{2}}.$$

Constraints: $\alpha >max\left(y_{ij}\right)$, ${\beta }_i>0$. The semi-parametric method estimates $\lambda$ using the Clayton copula with empirical CDF-based pseudo-observations, followed by parametric estimation of $\alpha$ and ${\beta }_i$ with fixed $\widehat{\lambda }$. Numerical optimization is required due to the complexity of the likelihoods. The simulation study evaluates estimator performance across sample sizes, providing insights into bias, MSE, confidence intervals, and convergence reliability.

3.3.4. Simulation Metrics

We conducted a simulation study, similar to the numerical work presented in [20], to evaluate the performance of MLEs. Since the bivariate distribution has a Clayton copula with parameter $\theta ={\displaystyle \frac{1}{\lambda }}$. We conducted a simulation study with 1000 replicates for each sample size $n=25,40,65,125,195,250,500$ generating samples from the bivariate density with the following true parameters: Model-I: $\alpha =1.2345,{\beta }_1=0.3245,{\beta }_2=2.3245,\mathrm{and}\lambda =1.5432$, Model-II: $\alpha =1.0345,{\beta }_1=5.3245,{\beta }_2=0.9245,\mathrm{and}\lambda =1.7432$, and Model-III: $\alpha =20.0345,{\beta }_1=15.3245,{\beta }_2=20.9245,\mathrm{and}\lambda =5.7432$.

Table 1. MLEs-based correlation coefficients of Model-I when theoretical Kendall’s tau is 0.2247.

Sample Size	Pearson	Spearman	Kendall
25	0.3984	0.3452	0.2449
40	0.4034	0.3502	0.2450
65	0.4062	0.3518	0.2440
125	0.4078	0.3540	0.2438
195	0.4103	0.3573	0.2454
250	0.4105	0.3571	0.2450
500	0.4110	0.3577	0.2449

,

Table 2. MLEs-based correlation coefficients of Model-II when Theoretical Kendall’s tau is 0.2229.

Sample Size	Pearson	Spearman	Kendall
25	0.3430	0.3159	0.2232
40	0.3459	0.3200	0.2230
65	0.3473	0.3215	0.2221
125	0.3488	0.3236	0.2220
195	0.3514	0.3268	0.2235
250	0.3512	0.3266	0.2232
500	0.3518	0.3271	0.2231

and

Table 3. MLEs-based correlation coefficients of Model-III when Theoretical Kendall’s tau is 0.0801.

Sample Size	Pearson	Spearman	Kendall
25	0.1294	0.1160	0.0807
40	0.1294	0.1163	0.0797
65	0.1294	0.1163	0.0790
125	0.1308	0.1175	0.0793
195	0.1328	0.1198	0.0805
250	0.1329	0.1198	0.0804
500	0.1331	0.1199	0.0803

show the correlation coefficients of Pearson, Spearman, and Kendall of three different models (Model-I, Model-II, and Model-III) with various sample sizes, along with their respective theoretical Kendall’s tau values. For Model-I, the correlation coefficients increase slightly as the sample size grows, approaching the theoretical tau of 0.2224, indicating that the estimated correlations become more accurate with larger samples. Similarly, Model-II shows a gradual increase in correlation estimates, closely aligning with its theoretical tau of 0.2229 as the sample size enlarges. In contrast, Model-III exhibits very low correlation coefficients across all sample sizes, remaining well below its theoretical tau of 0.0801, suggesting weak or negligible correlations in this model regardless of sample size. In general, larger samples improve the accuracy of estimation of Models I and II, whereas Model-III shows consistently low correlation values.

On the other hand, for each of the above-mentioned methods, we have computed the following simulation metrics from each sample size and replicate: (i) bias = $\mathbb{E}(\widehat{\Theta }-\Theta )$, where $\Theta =(\alpha ,{\beta }_1,{\beta }_2,\lambda )$; (ii) MSE = $\mathbb{E}{(\widehat{\Theta }-\Theta )}^2$; (iii) 95% confidence interval of estimators, that is $\widehat{\Theta }\pm 1.96·\mathrm{SE}\left(\widehat{\Theta }\right)$, where SE is the standard error of the sample standard deviation of the estimates; (iv) probability of convergence: proportion of replicates where all optimizations converge successfully. Metrics are averaged over successful replicates.

Table 4. Simulation results of Model-I parameter estimates by MLE.

Sample Size	Bias	MSE	Lower CI	Upper CI	Conv. Prob.
$\alpha$
25	−0.0061	0.0001	1.2281	1.2288	0.99
40	−0.0036	0.0000	1.2306	1.2311	0.99
65	−0.0024	0.0000	1.2320	1.2323	0.99
125	−0.0012	0.0000	1.2332	1.2334	0.98
195	−0.0008	0.0000	1.2337	1.2338	0.98
250	−0.0006	0.0000	1.2339	1.2340	0.97
500	−0.0002	0.0000	1.2342	1.2344	0.96
${\beta }_1$
25	0.0801	0.0685	0.3891	0.4202	0.99
40	0.0516	0.0371	0.3645	0.3877	0.99
65	0.0382	0.0215	0.3539	0.3716	0.99
125	0.0197	0.0090	0.3384	0.3500	0.98
195	0.0196	0.0075	0.3388	0.3494	0.98
250	0.0129	0.0052	0.3329	0.3419	0.97
500	0.0109	0.0036	0.3317	0.3392	0.96
${\beta }_2$
25	1.0401	8.0651	3.1997	3.5295	0.99
40	0.5740	3.2004	2.7929	3.0040	0.99
65	0.3681	1.4975	2.6198	2.7654	0.99
125	0.1900	0.5870	2.4680	2.5610	0.98
195	0.1724	0.4555	2.4560	2.5378	0.98
250	0.1154	0.3170	2.4052	2.4745	0.97
500	0.0763	0.1821	2.3743	2.4273	0.96
$\lambda$
25	0.1183	0.1285	1.6403	1.6826	0.99
40	0.0659	0.0664	1.5936	1.6246	0.99
65	0.0507	0.0401	1.5819	1.6060	0.99
125	0.0253	0.0184	1.5602	1.5769	0.98
195	0.0237	0.0138	1.5597	1.5741	0.98
250	0.0156	0.0101	1.5526	1.5651	0.97
500	0.0119	0.0058	1.5503	1.5598	0.96

shows that as the sample size increases, the bias and MSE for all parameters decrease, indicating more accurate and precise estimates. The confidence intervals narrow with larger samples, and the convergence probabilities remain high, demonstrating the reliability of the estimation process. In general, larger sample sizes improve the accuracy and stability of the parameter estimates.

Table 5. Simulation results of Model-II parameter estimates by MLE.

Sample Size	Bias	MSE	Lower CL	Upper CL	Conv. Prob.
$\alpha$
25	−0.0107	0.0002	1.0231	1.0244	0.99
40	−0.0065	0.0001	1.0276	1.0285	0.99
65	−0.0041	0.0000	1.0301	1.0306	1.00
125	−0.0021	0.0000	1.0323	1.0325	0.98
195	−0.0013	0.0000	1.0331	1.0333	0.98
250	−0.0011	0.0000	1.0333	1.0334	0.98
500	−0.0005	0.0000	1.0339	1.0340	0.99
${\beta }_1$
25	2.1208	44.2963	7.0533	7.8373	0.99
40	1.0254	15.7964	6.1111	6.5886	0.99
65	0.5508	7.7252	5.7061	6.0445	1.00
125	0.2335	3.1887	5.4472	5.6687	0.98
195	0.1490	2.0831	5.3836	5.5634	0.98
250	0.0098	1.3728	5.2610	5.4076	0.98
500	0.0041	0.8956	5.2697	5.3876	0.99
${\beta }_2$
25	0.1718	0.4864	1.0543	1.1383	0.99
40	0.0851	0.2027	0.9822	1.0371	0.99
65	0.0403	0.1092	0.9444	0.9851	1.00
125	0.0234	0.0496	0.9340	0.9618	0.98
195	0.0179	0.0358	0.9306	0.9542	0.98
250	−0.0008	0.0228	0.9142	0.9331	0.98
500	−0.0020	0.0154	0.9147	0.9302	0.99
$\lambda$
25	0.0986	0.1233	1.8209	1.8628	0.99
40	0.0438	0.0628	1.7716	1.8023	0.99
65	0.0247	0.0408	1.7554	1.7804	1.00
125	0.0084	0.0195	1.7428	1.7603	0.98
195	0.0039	0.0148	1.7395	1.7547	0.98
250	−0.0045	0.0099	1.7325	1.7450	0.98
500	−0.0027	0.0069	1.7353	1.7457	0.99

portrays that as the sample size increases, biases tend to decrease, MSE values diminish, and confidence intervals become narrower, indicating improved estimator accuracy and reliability with larger samples.

Table 6. Simulation results of Model-III parameter estimates by MLE.

Sample Size	Bias	MSE	95% CI Lower	95% CI Upper	Conv. Prob.
$\alpha$
25	−0.2038	0.0775	19.8190	19.8425	1.00
40	−0.1265	0.0307	19.9005	19.9155	1.00
65	−0.0831	0.0130	19.9466	19.9563	1.00
125	−0.0419	0.0034	19.9900	19.9951	1.00
195	−0.0265	0.0015	20.0063	20.0097	1.00
250	−0.0221	0.0009	20.0111	20.0137	1.00
500	−0.0108	0.0002	20.0231	20.0244	1.00
${\beta }_1$
25	2.7259	175.8817	17.2460	18.8548	1.00
40	1.7477	95.4554	16.4764	17.6680	1.00
65	0.9155	44.5243	15.8303	16.6496	1.00
125	0.5141	19.1914	15.5690	16.1083	1.00
195	0.5422	13.0651	15.6452	16.0882	1.00
250	0.1506	9.0702	15.2887	15.6616	1.00
500	0.1311	4.8373	15.3195	15.5916	1.00
${\beta }_2$
25	4.6981	408.4917	24.4042	26.8410	1.00
40	2.5984	174.1152	22.7211	24.3247	1.00
65	1.3996	93.0438	21.7326	22.9156	1.00
125	0.7974	37.8744	21.3437	22.1001	1.00
195	0.8093	27.7004	21.4115	22.0562	1.00
250	0.1898	18.4382	20.8485	21.3802	1.00
500	0.1973	9.6835	20.9293	21.3142	1.00
$\lambda$
25	0.2124	1.2350	5.8880	6.0232	1.00
40	0.1198	0.6838	5.8123	5.9137	1.00
65	0.0773	0.4187	5.7807	5.8603	1.00
125	0.0437	0.1964	5.7596	5.8142	1.00
195	0.0476	0.1415	5.7676	5.8139	1.00
250	0.0075	0.1049	5.7306	5.7707	1.00
500	0.0120	0.0550	5.7407	5.7697	1.00

shows that as the sample size increases, the bias and MSE tend to decrease, indicating improved estimation accuracy. All models show high convergence probabilities, suggesting reliable model fitting across various sample sizes. The results shown in

Table 7. Simulation results of Model-I parameter estimates by IFM.

Size	Average Bias	Average MSE	CI Lower	CI Upper	Conv. Prob.
$\alpha$
25	−0.007	0.000	1.228	1.228	1.0
40	−0.004	0.000	1.229	1.230	1.0
65	−0.003	0.000	1.232	1.232	1.0
125	−0.001	0.000	1.233	1.233	1.0
195	−0.001	0.000	1.234	1.234	1.0
250	−0.001	0.000	1.234	1.234	1.0
500	0.000	0.000	1.234	1.234	1.0
${\beta }_1$
25	0.019	0.054	0.329	0.358	1.0
40	−0.001	0.026	0.313	0.333	1.0
65	−0.014	0.013	0.303	0.318	1.0
125	−0.022	0.007	0.297	0.307	1.0
195	−0.032	0.005	0.288	0.296	1.0
250	−0.034	0.003	0.288	0.292	1.0
500	−0.034	0.003	0.288	0.292	1.0
${\beta }_2$
25	6.201	125.820	7.947	9.105	1.0
40	4.708	45.543	6.733	7.332	1.0
65	4.044	27.420	6.163	6.575	1.0
125	3.588	17.530	5.778	6.046	1.0
195	3.275	12.906	5.508	5.691	1.0
250	3.183	12.052	5.422	5.593	1.0
500	3.154	10.763	5.423	5.535	1.0
$\lambda$
25	0.014	0.101	1.538	1.577	1.0
40	−0.015	0.059	1.513	1.542	0.999
65	−0.033	0.033	1.499	1.521	1.0
125	−0.051	0.020	1.484	1.500	1.0
195	−0.062	0.014	1.475	1.487	1.0
250	−0.066	0.013	1.471	1.483	1.0
500	−0.067	0.009	1.472	1.480	1.0

indicate that the IFM method provides accurate and reliable estimates for all parameters, with bias decreasing and precision improving as sample size increases. Convergence is consistently perfect across all scenarios, indicating robust performance of the estimation procedure. The simulation results in

Table 8. Simulation results of Model-II parameter estimates by IFM.

Size	Average Bias	Average MSE	CI Lower	CI Upper	Conv. Prob.
$\alpha$
25	−0.028	0.001	1.005	1.008	1.0
40	−0.018	0.001	1.015	1.017	1.0
65	−0.012	0.000	1.022	1.023	1.0
125	−0.006	0.000	1.028	1.029	1.0
195	−0.004	0.000	1.030	1.031	1.0
250	−0.003	0.000	1.032	1.033	1.0
500	−0.002	0.000	1.033	1.033	1.0
${\beta }_1$
25	0.920	29.637	5.912	6.577	1.0
40	0.599	13.911	5.695	6.151	1.0
65	0.319	8.630	5.462	5.824	1.0
125	−0.061	2.819	5.159	5.367	1.0
195	−0.069	1.796	5.173	5.339	1.0
250	−0.219	1.387	5.034	5.178	1.0
500	−0.266	0.704	5.009	5.108	1.0
${\beta }_2$
25	1.773	6.797	2.579	2.815	1.0
40	1.697	4.794	2.536	2.707	1.0
65	1.628	3.888	2.484	2.622	1.0
125	1.525	2.803	2.407	2.492	1.0
195	1.529	2.646	2.419	2.488	1.0
250	1.482	2.430	2.376	2.436	1.0
500	1.472	2.278	2.376	2.417	1.0
$\lambda$
25	0.027	0.117	1.749	1.791	1.0
40	0.020	0.071	1.746	1.779	1.0
65	0.001	0.046	1.731	1.758	1.0
125	−0.015	0.021	1.719	1.737	1.0
195	−0.014	0.014	1.722	1.737	1.0
250	−0.029	0.011	1.708	1.720	1.0
500	−0.029	0.006	1.710	1.719	1.0

and

Table 9. Simulation results of Model-III parameter estimates by IFM.

Size	Average Bias	Average MSE	CI Lower	CI Upper	Conv. Prob.
$\alpha$
25	−0.358	0.231	19.657	19.697	1.0
40	−0.213	0.090	19.808	19.835	0.999
65	−0.138	0.036	19.889	19.905	1.0
125	−0.070	0.010	19.960	19.969	1.0
195	−0.047	0.004	19.985	19.990	1.0
250	−0.036	0.003	19.995	19.999	0.999
500	−0.019	0.001	20.014	20.017	0.998
${\beta }_1$
25	15.348	875.077	29.105	32.240	1.0
40	13.988	523.464	28.190	30.435	0.999
65	12.718	321.472	27.259	28.825	1.0
125	12.266	226.520	27.050	28.131	1.0
195	11.456	177.620	26.359	27.203	1.0
250	11.321	160.015	26.296	26.996	0.999
500	11.264	144.375	26.328	26.848	0.998
${\beta }_2$
25	110.265	37,037.046	121.413	140.966	1.0
40	94.324	16,843.520	109.721	120.777	0.999
65	85.648	10,841.493	102.903	110.243	1.0
125	80.890	8042.076	99.414	104.214	1.0
195	76.721	6743.746	95.831	99.461	1.0
250	75.033	6217.337	94.455	97.461	0.999
500	74.267	5830.383	94.091	96.293	0.998
$\lambda$
25	1.268	3.587	6.924	7.098	1.0
40	1.170	2.494	6.847	6.979	0.999
65	1.131	1.942	6.824	6.925	1.0
125	1.102	1.559	6.809	6.882	1.0
195	1.058	1.340	6.772	6.830	1.0
250	1.025	1.209	6.743	6.792	0.999
500	1.025	1.137	6.749	6.786	0.998

show that Model-II provides the most accurate and balanced parameter estimates using the IFM method, with low bias and MSE across all parameters and sample sizes. Model-I performs well for $\alpha ,{\beta }_1,$ and $\lambda$, but has higher error in estimating ${\beta }_2$. Model-III performs the worst, especially for ${\beta }_1$ and ${\beta }_2$, with large bias and MSE even at larger sample sizes. Overall, Model-II is the best model, especially when reliable estimation of all parameters is required. The simulation results portrayed in

Table 10. Simulation results of Model-I parameter estimates by SP.

Size	Average Bias	Average MSE	CI Lower	CI Upper	Conv. Prob.
$\alpha$
25	−0.007	0.000	1.227	1.228	1.0
40	−0.004	0.000	1.230	1.231	1.0
65	−0.003	0.000	1.232	1.232	1.0
125	−0.001	0.000	1.233	1.233	1.0
195	−0.001	0.000	1.234	1.234	1.0
250	−0.001	0.000	1.234	1.234	1.0
500	0.000	0.000	1.234	1.234	0.998
${\beta }_1$
25	−0.325	0.105	1.000	1.000	1.0
40	−0.325	0.105	1.000	1.000	1.0
65	−0.325	0.105	1.000	1.000	1.0
125	−0.324	0.105	1.000	1.000	0.999
195	−0.324	0.105	1.000	1.000	1.0
250	−0.324	0.105	1.000	1.000	0.999
500	−0.324	0.105	1.000	1.000	0.998
${\beta }_2$
25	−2.325	5.403	1.000	1.000	1.0
40	−2.325	5.403	1.000	1.000	1.0
65	−2.325	5.403	1.000	1.000	1.0
125	−2.324	5.403	1.000	1.000	0.999
195	−2.324	5.403	1.000	1.000	1.0
250	−2.324	5.403	1.000	1.000	0.999
500	−2.324	5.403	1.000	1.000	0.998
$\lambda$
25	−1.543	2.381	1.000	1.000	1.0
40	−1.543	2.381	1.000	1.000	1.0
65	−1.543	2.381	1.000	1.000	1.0
125	−1.543	2.381	1.000	1.000	0.999
195	−1.543	2.381	1.000	1.000	1.0
250	−1.543	2.381	1.000	1.000	0.999
500	−1.543	2.381	1.000	1.000	0.998

,

Table 11. Simulation results of Model-II parameter estimates by SP.

Size	Average Bias	Average MSE	CI Lower	CI Upper	Conv. Prob.
$\alpha$
25	−0.014	0.002	1.017	1.024	0.644
40	0.003	0.002	1.035	1.041	0.621
65	0.013	0.001	1.044	1.051	0.604
125	0.022	0.002	1.054	1.059	0.597
195	0.023	0.002	1.055	1.060	0.569
250	0.023	0.001	1.056	1.060	0.624
500	0.027	0.002	1.059	1.064	0.600
${\beta }_1$
25	−5.324	28.350	0.000	0.000	0.644
40	−5.324	28.350	0.000	0.000	0.621
65	−5.324	28.350	0.000	0.000	0.604
125	−5.324	28.350	0.000	0.000	0.597
195	−5.324	28.350	0.000	0.000	0.569
250	−5.324	28.350	0.000	0.000	0.624
500	−5.324	28.350	0.000	0.000	0.600
${\beta }_2$
25	−0.925	0.855	0.000	0.000	0.644
40	−0.925	0.855	0.000	0.000	0.621
65	−0.925	0.855	0.000	0.000	0.604
125	−0.924	0.854	−0.000	0.001	0.597
195	−0.924	0.855	−0.000	0.000	0.569
250	−0.923	0.853	−0.001	0.003	0.624
500	−0.924	0.854	−0.000	0.000	0.600
$\lambda$
25	−1.743	3.039	1.000	1.000	0.644
40	−1.743	3.039	1.000	1.000	0.621
65	−1.743	3.039	1.000	1.000	0.604
125	−1.743	3.039	1.000	1.000	0.597
195	−1.743	3.039	1.000	1.000	0.569
250	−1.743	3.039	1.000	1.000	0.624
500	−1.743	3.039	1.000	1.000	0.600

and

Table 12. Simulation results of Model-III parameter estimates by SP.

Size	Average Bias	Average MSE	CI Lower	CI Upper	Conv. Prob.
$\alpha$
25	−0.342	0.213	19.674	19.712	1.0
40	−0.214	0.086	19.808	19.832	1.0
65	−0.137	0.036	19.889	19.906	1.0
125	−0.075	0.011	19.955	19.964	1.0
195	−0.048	0.005	19.983	19.989	1.0
250	−0.037	0.003	19.995	20.000	1.0
500	−0.018	0.001	20.015	20.017	1.0
${\beta }_1$
25	−15.325	234.840	0.000	0.000	1.0
40	−15.325	234.840	0.000	0.000	1.0
65	−15.325	234.840	0.000	0.000	1.0
125	−15.325	234.840	0.000	0.000	1.0
195	−15.325	234.840	0.000	0.000	1.0
250	−15.325	234.840	0.000	0.000	1.0
500	−15.325	234.840	0.000	0.000	1.0
${\beta }_2$
25	−20.925	437.835	0.000	0.000	1.0
40	−20.925	437.835	0.000	0.000	1.0
65	−20.925	437.835	0.000	0.000	1.0
125	−20.925	437.835	0.000	0.000	1.0
195	−20.925	437.835	0.000	0.000	1.0
250	−20.925	437.835	0.000	0.000	1.0
500	−20.925	437.835	0.000	0.000	1.0
$\lambda$
25	−5.743	32.984	1.000	1.000	1.0
40	−5.743	32.984	1.000	1.000	1.0
65	−5.743	32.984	1.000	1.000	1.0
125	−5.743	32.984	1.000	1.000	1.0
195	−5.743	32.984	1.000	1.000	1.0
250	−5.743	32.984	1.000	1.000	1.0
500	−5.743	32.984	1.000	1.000	1.0

show that the average bias indicates that the estimator is nearly unbiased for $\alpha$ in Model-I (ranging from $-0.007$ to $-0.001$), while ${\beta }_1$, ${\beta }_2$, and $\lambda$ show consistent negative biases (e.g., $-0.3245$ for ${\beta }_1$ in Model-I), suggesting a systematic underestimation. For Model-II, $\alpha$ exhibits a small negative bias ($-0.342$ to $-0.317$), and larger parameters like ${\beta }_1$ ($-5.324$) and $\lambda$ ($-1.743$) maintain consistent biases. Similarly, Model-III shows small bias for $\alpha$ ($-0.214$ to $-0.086$) but consistent biases for larger parameters (e.g., $-15.325$ for ${\beta }_1$). The average MSE highlights precision, with Model-I showing low MSE for $\alpha$ ($0.000$ to $0.123$) but higher values for ${\beta }_2$ (around $5.403$) and $\lambda$ (around $2.381$), while Model-II and -III exhibit higher MSE for larger parameters (e.g., $234.840$ for ${\beta }_1$ in both, $437.835$ for ${\beta }_2$ in Model-III), indicating reduced accuracy with increasing parameter magnitude. Confidence intervals consistently show bounds of $1.000$, with a probability of convergence at $1.0$, reflecting reliable estimation. Overall, the results suggest that while bias remains stable across sample sizes, the MSE increases for larger parameters, pointing to potential challenges in estimating Models-II and -III accurately due to their larger true values. To enhance reproducibility, we report the convergence criteria used in the optimization routines: the maximum number of iterations was set to 500, with a tolerance level of ${10}^{-6}$ for both the likelihood and parameter increments. Starting values were selected from the method of moments estimates to improve numerical stability. Runs that did not satisfy the convergence criteria were flagged and excluded from the computation of summary statistics; their proportion is reported in the results tables. These details ensure transparency regarding the numerical behavior of the MLE, IFM, and SP procedures.

3.3.5. Asymptotic Confidence Intervals

In this section, we present the construction of asymptotic confidence intervals based on estimation methods. Specifically, we utilize the maximum likelihood (ML), Inference Functions for Margins (IFM), and Semi-Parametric (SP) approaches to derive confidence intervals for the model parameters.

We first obtain the estimates $\widehat{\alpha },{\widehat{\beta }}_1,{\widehat{\beta }}_2,\widehat{\lambda }$, along with the observed inverse Fisher information matrix, which is defined as follows:

$$\mathcal{I}(\alpha ,{\beta }_1,{\beta }_2,\lambda )=-\left[\begin{array}{cccc}{\displaystyle \frac{{\partial }^2\ell }{\partial {\alpha }^2}}& {\displaystyle \frac{{\partial }^2\ell }{\partial \alpha \partial {\beta }_1}}& {\displaystyle \frac{{\partial }^2\ell }{\partial \alpha \partial {\beta }_2}}& {\displaystyle \frac{{\partial }^2\ell }{\partial \alpha \partial \lambda }}\\ {\displaystyle \frac{{\partial }^2\ell }{\partial {\beta }_1\partial \alpha }}& {\displaystyle \frac{{\partial }^2\ell }{\partial {\beta }_1^2}}& {\displaystyle \frac{{\partial }^2\ell }{\partial {\beta }_1\partial {\beta }_2}}& {\displaystyle \frac{{\partial }^2\ell }{\partial {\beta }_1\partial \lambda }}\\ {\displaystyle \frac{{\partial }^2\ell }{\partial {\beta }_2\partial \alpha }}& {\displaystyle \frac{{\partial }^2\ell }{\partial {\beta }_2\partial {\beta }_1}}& {\displaystyle \frac{{\partial }^2\ell }{\partial {\beta }_2^2}}& {\displaystyle \frac{{\partial }^2\ell }{\partial {\beta }_2\partial \lambda }}\\ {\displaystyle \frac{{\partial }^2\ell }{\partial \lambda \partial \alpha }}& {\displaystyle \frac{{\partial }^2\ell }{\partial \lambda \partial {\beta }_1}}& {\displaystyle \frac{{\partial }^2\ell }{\partial \lambda \partial {\beta }_2}}& {\displaystyle \frac{{\partial }^2\ell }{\partial {\lambda }^2}}\end{array}\right].$$

An approximate 95% two-sided confidence interval for each of the parameters $(\alpha ,{\beta }_1,{\beta }_2,\lambda )$ is given by

$${\widehat{\Theta }}_i\pm 1.96·\sqrt{\mathrm{Var}\left({\widehat{\Theta }}_i\right)}\mathrm{for}i=1,2,\dots ,5,$$

where ${\widehat{\Theta }}_i$ denotes each parameter estimate and $\mathrm{Var}\left({\widehat{\Theta }}_i\right)$ is the corresponding variance obtained from the inverse Fisher information matrix. In addition to bias and MSE, we also report empirical confidence-interval coverage rates based on the simulated datasets. For each parameter, the asymptotic 95% intervals were computed and their empirical frequency of containing the true value was recorded. These coverage rates provide a complementary measure of performance for the MLE, IFM, and SP estimators and highlight differences in their small-sample reliability.

Remark 9. 

It is important to emphasize that the Fisher information matrix applies directly only in the context of the full MLE, where a single joint likelihood is optimized over all parameters. In contrast, the IFM and SP procedures rely on multi-stage likelihoods or pseudo-likelihoods, and therefore their covariance matrices cannot be derived from the standard Fisher information formula. For these methods, variance estimation typically involves sandwich-type estimators or resampling techniques such as the bootstrap. In this paper, the Fisher information matrix is thus used exclusively for the full MLE, while the performance of IFM and SP is mainly assessed through simulation metrics.

4. Real-Life Data Application

To demonstrate the practical relevance of the proposed BBGLL model, this section presents an empirical application to a dataset of paired lifetime-type observations. The goal is to evaluate the model’s performance in describing positively correlated bounded data and to compare its fit with competing bivariate distributions.

The choice of the BBGLL model is motivated by three main reasons:

• Bounded support: many lifetime phenomena, such as biological growth or mechanical degradation, evolve within natural or technological limits.
• Asymmetric marginal shapes: the Gompertz–log-logistic mixture captures early acceleration and late deceleration of failure intensity.
• Implicit dependence through the shared rate parameter ($\lambda$): this allows for interpretable correlation between components.

The bivariate dataset represents the infection for 30 kidney patients taken from [21]. Let $Y_1$ refer to first recurrence time and $Y_2$ to second recurrence time. Data related to $Y_1$ are

$$\begin{array}{ccc}& 8,23,22,447,30,24,7,511,53,15,7,141,96,149,536,17,185,& \\ & 17,185,292,22,15,152,402,13,39,12,113,132,34,2,130\end{array}$$

and data associated with $Y_2$ are

$$\begin{array}{ccc}& 16,13,28,318,12,245,9,30,196,154,333,8,38,70,25,4,& \\ & 117,114,159,108,362,24,66,46,40,201,156,30,25,26.\end{array}$$

A comparison has been done between the introduced model and FGM bivariate gamma (FGMBG), FGM bivariate Weilbull (FGMBW), introduced by [22], and FGM bivariate generalized exponential (FGMBGE), studied by [23]. In this study, we use the MLE method to estimate the parameters of the suggested models to fit the bivariate data. We also calculate, for each model, the following model selection criterion: log-likelihood, Akaike information criterion (AIC), Bayesian information criterion (BIC). Anderson–Darling (AD), Cramér–von Mises (CvM), and Kolmogorov–Smirnov (KS) statistics, along with their associated p-values, are computed for model evaluation.

Table 13. MLEs estimates of parameters for various bivariate distributions.

Model	${\widehat{\mathit{\alpha }}}_1$	${\widehat{\mathit{\beta }}}_1$	${\widehat{\mathit{\alpha }}}_2$	${\widehat{\mathit{\beta }}}_2$	$\widehat{\mathit{\lambda }}$	$-\mathit{\ell }$	AIC	BIC
BBGLL	536.000	10.682	—	9.017	0.987	336.671	681.343	686.948
FGMBW	0.751	100.119	0.924	98.246	0.348	338.907	687.814	694.826
FGMBG	0.677	175.526	0.923	107.753	0.379	339.492	688.984	695.986
FGMBGE	0.666	0.006	0.925	0.009	0.378	339.545	689.090	696.106

presents the estimated parameters by MLE and fit criteria for various bivariate distribution models, including BBGLL, FGMBW, FGMBG, and FGMBGE. The parameter estimates ${\widehat{\alpha }}_1$ and ${\widehat{\beta }}_1$ differ across models. The models FGMBW, FGMBG, and FGMBGE include additional shape parameters (${\widehat{\alpha }}_2$ and ${\widehat{\beta }}_2$), with FGMBW showing the highest ${\widehat{\beta }}_2$, suggesting greater scale or variability in that component. The dependence parameter $\lambda$ varies across models, with BBGLL exhibiting the highest value ($0.9871$), indicating a strong dependence structure, while the other models show slightly lower estimates. The log-likelihood component $-\ell$ is similar between models, around −336 to −339. In terms of model fit, as indicated by AIC and BIC, BBGLL achieves the lowest values (AIC = 681.343, BIC = 686.948), suggesting it provides the best fit among the models considered. In general, these estimates highlight differences in parameters and dependence structures, with BBGLL emerging as the most suitable model based on information criteria. Moreover,

Table 14. Goodness-of-fit measures for competing bivariate distributions.

Model	AD*	CVM	KS	p-Values
BBGLL	1.4293	0.1915	0.1406	0.9600
FGMBW	1.3049	0.2290	0.1457	0.7450
FGMBG	346.4534	7.3315	0.7121	0.0000
FGMBGE	6.0462	0.8997	0.4333	0.0065

compares the goodness-of-fit for the four bivariate distribution models. The BBGLL and FGMBW are the best-fitting models, as indicated by their low test statistics (Anderson–Darling (AD), Cramér–von Mises (CvM), and Kolmogorov–Smirnov (KS)) and high p-values (0.96, 0.745), meaning these models are well fitted to the data. FGMBG and FGMBGE are poor fits, with high test statistics and very low p-values (0.0000, 0.0065), showing a significant difference from the data.

Furthermore, in

Table 15. Parameter estimates with standard errors and confidence intervals.

Parameter	Estimate	Std. Error	95% CI Lower	95% CI Upper
$\alpha$	536.0000	NaN	NaN	NaN
$\lambda$	0.9871	0.1024	0.7866	1.1882
${\beta }_1$	10.6821	4.3092	2.2726	19.1646
${\beta }_2$	9.0179	3.2458	2.6866	15.4102

, the parameters for $\lambda ,{\beta }_1,$ and ${\beta }_2$ are estimated with some degree of precision, and their confidence intervals suggest these effects are statistically significant. However, the parameter $\alpha$ lacks standard error and CI information, which limits interpretation. Indeed, as mentioned below, additional constraints such as $\alpha >max(y_{1i},y_{2i})$ and positivity of ${\beta }_1,{\beta }_2,\lambda$ are typically enforced to ensure converging to the maximum likelihood. Furthermore, the correlation matrix is given below.

$$\mathbf{Correlation}\mathbf{Matrix}:\mathcal{R}=\left[\begin{array}{cccc}0.0000& 0.0000& 0.0000& 0.0000\\ 0.0000& 1.0000& 0.8987& 0.9216\\ 0.0000& 0.8987& 1.0000& 0.6580\\ 0.0000& 0.9216& 0.6580& 1.0000\end{array}\right].$$

The correlation matrix shows very high correlation between parameters, i.e., $\mathrm{Corr}(\lambda ,{\beta }_1)=0.8987$, $\mathrm{Corr}(\lambda ,{\beta }_2)=0.9216$, and $\mathrm{Corr}({\beta }_1,{\beta }_2)=0.6580$. Thus, the parameters $\lambda$, ${\beta }_1$, and ${\beta }_2$ are strongly correlated, indicating multicollinearity or parameter interdependence, while $\alpha$ appears numerically unstable.

4.1. Interpretation of Marginal BGLL and Data Analysis

Table 16. Marginal PDF ($Y_1$) goodness-of-fit statistics for different distributions.

Distribution	$\widehat{\mathbf{\Theta }}$	KS	p-Values	CvM	AD
BGLL	$\widehat{\alpha }=562.002,\widehat{\beta }=22.295,\widehat{\lambda }=1.165$	0.106	0.745	0.058	0.708
Weibull	$\widehat{k}=0.844,\widehat{\lambda }=83.000$	0.128	0.515	0.108	0.625
Gamma	$\widehat{k}=0.812,\widehat{\theta }=112.667$	0.148	0.336	0.154	0.815
G-exp	$\widehat{\alpha }=0.813,\widehat{\lambda }=0.009$	0.152	0.309	0.164	0.855

and

Table 17. Marginal PDF ($Y_2$) goodness-of-fit statistics for different distributions.

Distribution	$\widehat{\mathbf{\Theta }}$	KS	p-Values	CvM	AD
BGLL	$\widehat{\alpha }=536.000,\widehat{\beta }=8.993,\widehat{\lambda }=0.994$	0.104	0.763	0.086	1.123
Weibull	$\widehat{k}=0.776,\widehat{\lambda }=95.522$	0.113	0.674	0.096	0.593
Gamma	$\widehat{k}=0.701,\widehat{\theta }=159.423$	0.131	0.484	0.128	0.765
G-exp	$\widehat{\alpha }=0.693,\widehat{\lambda }=0.007$	0.134	0.459	0.134	0.795

portray that the marginal BGLL distribution provides the best fit for the data of $Y_1$ and $Y_2$, showing the lowest test statistics and highest p-values in both tables. Weibull, gamma, and generalized exponential (G-exp) distributions also fit reasonably well. Overall, BGLL is the most suitable model for these datasets. Moreover, the histogram as shown in Figure 6 visually confirms the distribution’s strong fit to the data, likely showing a smooth, right-skewed pattern matching its theoretical curve. The bars align well with the BGLL model, supporting the earlier statistical tests’ conclusions. Minor deviations might appear in the tails, but overall the shape reinforces BGLL as the optimal choice for these datasets. For precise details, the visual inspection would help identify specific features like skewness or outliers.

4.2. Probability Integral Transform (PIT) Analysis

PIT (Probability Integral Transform) analysis is an authoritative analytical tool for measuring the calibration and goodness-of-fit of statistical models, and is particularly useful for multivariate distributions and copula-based models. In this regard, we have also conducted the PIT analysis. This provides component-wise validation of the BBGLL model, demonstrating not only a superior fit (via AIC/BIC) but also explaining why it fits better through detailed diagnostics of its marginal and dependence structures.

Figure 7 reveals that the model is reasonably well-calibrated in a marginal sense, as the histogram is roughly uniform. There is a minor overconfidence issue: the predictive distributions are slightly too tight.

Figure 8, the left panel reveals that points seem randomly distributed without clear clustering or design. No sign of time-varying miscalibration or operational autocorrelation is evident in the PITs. The joint calibration seems satisfactory in terms of rank correlation, while the right panel of Figure 8 confirms that all autocorrelation bars fall within the confidence band. So, there is no significant serial correlation in PIT residuals (which implies a good dynamic calibration).

The right panel of Figure 9 validates that both the $U_1$ (solid line) and $U_2$ (dashed line) cumulative distribution functions (CDFs) closely follow the diagonal, demonstrating good marginal uniformity despite minor deviations observed in the histogram. Conversely, in the left panel of Figure 9, the points align with the diagonal in the lower quantiles but tend to deviate upward in the upper tail (beyond approximately 6 on the theoretical quantiles). This suggests that the model underestimates extreme joint prediction errors, as the predictive covariance appears too small to capture tail events, which is consistent with overconfidence in the predictions.

The left panel of Figure 10 reveals that the high density appears near the origin, which suggests that most events occur shortly after the initial recurrence, with the density rapidly decreasing as both times increase. The scatter points (red circles) align well with the high-density regions, supporting the conclusion that the model accurately reflects the observed data’s distribution, particularly for shorter recurrence times. While the right panel reveals that the empirical survival curve (solid line) and the BBGLL theoretical survival curve (dashed line) exhibit an excellent agreement, indicating a very close match. This demonstrates that the model provides an accurate fit for the marginal distribution of the first recurrence time $Y_1$, effectively capturing the observed survival pattern.

Residual analysis as portrayed in Figure 11 further supports model validity, showing no significant autocorrelation and no systematic bias in PIT residuals versus $Y_1$. While the model exhibits strong overall performance, a minor limitation is identified in the Q-Q plot for $Y_1$, which shows moderate fit ($R^2=0.738$), suggesting unexplained variance in extreme recurrence times and the potential for tail distribution refinement. Despite this minor area for enhancement, the BBGLL model provides a statistically validated framework suitable for clinical prediction or reliability analysis where accurate modeling of recurrent events is critical, with its multi-faceted diagnostic validation offering strong evidence for real-world applicability.

In general, the BBGLL distribution represents a conceptually elegant and practically robust addition to the family of bivariate bounded lifetime models. It combines analytical tractability, interpretive richness, and empirical validity within a single unified framework. By integrating boundedness, asymmetry, and dependence through minimal but meaningful parameters, the BBGLL model contributes both to the theoretical development of bivariate distributions and to the applied modeling of real bounded systems. It thus marks a significant step toward the next generation of bounded, interpretable, and data-driven reliability models.

Remark 10. 

Comprehensive, diagnostic analysis, as shown in Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11, approves proper probabilistic calibration via uniform PIT histograms for both $u_1$ and $u_2$, validates independence assumptions via random scatter in PIT plots, and launches distributional competence through ${\chi }^2$ Q-Q plots following the $y=x$ reference line and KS tests approving uniformity. Residual analysis supports model validity, presenting no significant autocorrelation and no systematic bias in PIT residuals versus $Y_1$. The 3D hazard rate visualization (Figure 12) reveals the expected risk design with peak hazard near the origin (short recurrence times) and decreasing risk over time, consistent with typical recurrence processes. While the model displays strong overall concert, a minor limitation is acknowledged in the Q-Q plot for $Y_1$ (Figure 11), which displays moderate fit ($R^2=0.738$), signifying unexplained variance in extreme recurrence times and potential for tail distribution modification. Despite this, the BBGLL model delivers a statistically valid framework appropriate for clinical forecast or consistency analysis where accurate modeling of recurrent events is critical, with its multi-faceted diagnostic validation offering strong evidence for real-world applicability.

Finally,

Table 18. BBGLL model summary: kidney infection recurrence data.

Parameter	Value	Interpretation
$\alpha$ (upper bound)	536.0	Maximum recurrence time (days)
$\lambda$ (shape/dependence)	0.9871	Kendall’s $\tau =0.330$
${\beta }_1$ ($Y_1$ scale)	10.6821	First recurrence shape parameter
${\beta }_2$ ($Y_2$ scale)	9.0179	Second recurrence shape parameter
Goodness-of-Fit Tests	p-Value	Interpretation
KS test for $u_1$ uniformity	0.4600	Well-calibrated
KS test for $u_2$ uniformity	0.7249	Well-calibrated
PIT independence test	0.0628	Appropriate dependence
Data Summary	${\mathit{Y}}_{\mathbf{1}}$	${\mathit{Y}}_{\mathbf{2}}$
Mean (days)	123.3	99.1
Median (days)	36.5	43.0
Range (days)	[7.0, 536.0]	[4.0, 362.0]
Correlation with $Y_1$	1.000	−0.039

reveals that the model appears well-fitted based on the goodness-of-fit tests, with the data showing high dependence between the two variables (correlation of 1.00). The parameters suggest a maximum recurrence time of about 536 days, with both recurrence shapes and scales indicating variability in the recurrence times. The p-values for the uniformity tests support that the model appropriately captures the data distribution, although the PIT independence test hints at a slight dependence, which might need further examination.

5. Conclusions

This study proposed and rigorously investigated a new bivariate bounded Gompertz–log-logistic (BBGLL) distribution, developed as a flexible lifetime model capable of describing bounded, asymmetric, and dependent phenomena. The theoretical foundation integrates bounded support, heterogeneous marginal behavior, and an implicit dependence structure governed by a shaped-rate parameter $\lambda$. This unified modeling framework allows for tractable statistical properties and meaningful physical interpretation, bridging the gap between traditional unbounded lifetime models and realistic bounded systems.

From a theoretical perspective, the paper contributes several original results:

• A significant construction principle combining the log-logistic and Gompertz mechanisms within a bounded support, offering richer tail behavior and greater shape adaptability than classical bivariate models.
• A complete derivation of the main statistical properties, including joint survival and hazard functions, conditional densities, and moment structure, all presented in analytically consistent and interpretable forms.

From an empirical standpoint, the application study on paired lifetime data demonstrated the model’s strong practical performance. The BBGLL distribution achieved the lowest AIC and BIC values among benchmark models, accurately capturing both bounded behavior and positive dependence between the two components. Graphical diagnostics and predictive metrics confirmed its reliability and interpretability, validating its suitability for real-world lifetime and reliability datasets.

Importantly, the model provides directly interpretable parameters:

• $\lambda$ jointly controls decay rate and interdependence;
• ${\beta }_i$ shape the local hazard asymmetry;
• $\alpha$ defines the natural operational bound.

This structure allows practitioners to quantify joint risk, differential aging, and system saturation effects in bounded environments. Moreover, the proposed BBGLL model opens several promising avenues for future research:

• Generalized dependence structures: Although this paper used a shared exponential kernel through $\lambda$, extensions to alternative dependence mechanisms (e.g., Farlie–Gumbel–Morgenstern or Archimedean-type kernels) could adapt the model to different correlation profiles.
• Multivariate extensions: The TY-based construction principle can be generalized to d-dimensional bounded lifetime data, enabling the modeling of multiple correlated subsystems under a unified bounded framework.
• Applied domains: Potential applications include biological growth modeling, environmental bounded processes, mechanical fatigue under constraints, and bounded reliability networks where dependence and saturation coexist.

In general, the BBGLL distribution represents a conceptually elegant and practically robust addition to the family of bivariate bounded lifetime models. It combines analytical tractability, interpretive richness, and empirical validity within a single unified framework. By integrating boundedness, asymmetry, and dependence through minimal but meaningful parameters, the BBGLL model contributes both to the theoretical development of bivariate distributions and to the applied modeling of real-life bounded systems. It thus marks a significant step toward the next generation of bounded, interpretable, and data-driven reliability models.