Further results on the bivariate semi-parametric singular family of distributions

Abstract

General classes of bivariate distributions are well-studied in the literature. Most of these classes are proposed via a copula formulation or extensions of some characterization properties in the univariate case. In Kundu (2022), we see one such semi-parametric family useful to model bivariate data with ties. This model is a general semi-parametric model with a baseline. In this paper, we present a characterization property of this class of distributions in terms of a functional equation. The general solution to this equation is explored. Necessary and sufficient conditions under which the solution becomes a bivariate distribution are investigated. An application of the characterization property of the proposed class for generating bivariate pairs of random variables from a member distribution is also discussed.

Keywords

Bivariate distributions functional equation proportional hazard models semi-parametric class

1. Introduction

Analyzing bivariate datasets is very challenging as it involves the association between two variables. Data with ties are also common. The Marshall-Olkin copula, defined by

$\displaystyle C(u_{1},u_{2})=\left\{\begin{array}[]{ll}u_{1}^{1-p}u_{2};&u_{1}% ^{p}\geqslant u_{2}^{q}\\ u_{1}u_{2}^{1-q};&u_{1}^{p}\leqslant u_{2}^{q}\end{array}\right.$ (1)

where $0<p,q<1$ and $0<u_{i}<1;i=1,2$ is apt in modelling data with ties (see Nelsen (2007)). Many authors have proposed general classes of bivariate distributions and studied their properties. In Eq. (1), when $u_{i}$ is exponential ( $\theta_{i}+\theta_{3}$ ) for $i=1,2$ and $p=\frac{\theta_{3}}{\theta_{1}+\theta_{3}}$ and $q=\frac{\theta_{3}}{\theta_{2}+\theta_{3}}$ , the copula reduces to the well-studied Marshall-Olkin bivariate exponential distribution with a singular component and is specified as

$\displaystyle\bar{F}(x_{1},x_{2})=\left\{\begin{array}[]{ll}e^{-(\theta_{1}+% \theta_{3})x_{1}-\theta_{2}x_{2}};&x_{1}\geqslant x_{2}\\ e^{-\theta_{1}x_{1}-(\theta_{2}+\theta_{3})x_{2}};&x_{1}\leqslant x_{2},\end{% array}\right.$ (2)

where $\theta_{1},\theta_{2},\theta_{3}>0$ . Here $(X_{1},X_{2})$ is a bivariate random vector having survival function $\bar{F}(x_{1},x_{2})=P(X_{1}>x_{1},X_{2}>x_{2})$ with support $I_{(0,\infty)\times(0,\infty)}=\{(x_{1},x_{2});0<x_{1}<\infty,0<x_{2}<\infty\}$ . The distribution in Eq. (2) is characterized by the functional equation popularly known as bivariate lack of memory property (BLMP) (Marshall & Olkin, 1967) given by

$\displaystyle\bar{F}(x_{1}+t,x_{2}+t)=\bar{F}(x_{1},x_{2})\bar{F}(t,t)$

for all $x_{1},x_{2},t\geqslant 0$ . Kolev and Pinto (2018b) worked on Marshall-Olkin’s bivariate exponential distribution providing a weak version of the BLMP, which can be used to construct bivariate distributions having a singularity component along an arbitrary line through the origin. Lin et al. (2019) studied the moment generating function, product moments and dependence structure of the bivariate distributions satisfying the BLMP. Many authors have studied the BLMP from different perspectives. See Pinto and Kolev (2015) and Kolev and Pinto (2018a) to mention a few. In fact, the BLMP can be translated as a property of arbitrary bivariate continuous distributions (Galambos & Kotz, 2006).

If $u_{i}$ is Weibull ( $\theta_{i}+\theta_{3},\alpha$ ) for $i=1,2$ , the Marshall-Olkin copula in Eq. (1) gives a bivariate distribution with a singular component whose survival function is

$\displaystyle\bar{F}(x_{1},x_{2})=\left\{\begin{array}[]{ll}e^{-(\theta_{1}+% \theta_{3})x_{1}^{\alpha}-\theta_{2}x_{2}^{\alpha}};&x_{1}\geqslant x_{2}\\ e^{-\theta_{1}x_{1}^{\alpha}-(\theta_{2}+\theta_{3})x_{2}^{\alpha}};&x_{1}% \leqslant x_{2},\end{array}\right.$ (3)

where $\alpha,\theta_{1},\theta_{2},\theta_{3}>0$ , characterized by the functional equation,

$\displaystyle\bar{F}((x_{1}^{\alpha}+t^{\alpha})^{\frac{1}{\alpha}},(x_{2}^{% \alpha}+t^{\alpha})^{\frac{1}{\alpha}})=\bar{F}(x_{1},x_{2})\bar{F}(t,t),$

for all $x_{1},x_{2},t\geqslant 0$ .

If $u_{i}$ is the one-parameter Pareto ( $\theta_{i}+\theta_{3}$ ) for $i=1,2$ , the Marshall-Olkin copula in Eq. (1) gives a bivariate distribution with a singular component whose survival function is

$\displaystyle\bar{F}(x_{1},x_{2})=\left\{\begin{array}[]{ll}\left(\frac{1}{x_{% 1}}\right)^{\theta_{1}+\theta_{3}}\left(\frac{1}{x_{2}}\right)^{\theta_{2}};&x% _{1}\geqslant x_{2}\\ \left(\frac{1}{x_{1}}\right)^{\theta_{1}}\left(\frac{1}{x_{2}}\right)^{\theta_% {2}+\theta_{3}};&x_{1}\leqslant x_{2},\end{array}\right.$ (4)

where $\theta_{1},\theta_{2},\theta_{3}>0$ , characterized by the functional equation,

$\displaystyle\bar{F}(x_{1}t,x_{2}t)=\bar{F}(x_{1},x_{2})\bar{F}(t,t),$

for all $x_{1},x_{2},t\geqslant 0$ .

If the marginals belong to the proportional hazard (PH) class given by $u_{i}=\bar{F}_{i}(x_{i})=[\bar{F}_{0}(x_{i})]^{\theta_{i}+\theta_{3}}$ for $i=1,2$ , then Eq. (1) becomes the copula associated with the bivariate proportional hazard class (BPHC) mentioned in Kundu (2022), the survival function of which is given by

$\displaystyle\bar{F}(x_{1},x_{2})=\left\{\begin{array}[]{ll}{[}\bar{F}_{0}(x_{% 1})]^{\theta_{1}+\theta_{3}}[\bar{F}_{0}(x_{2})]^{\theta_{2}};&x_{1}\geqslant x% _{2}\\ {[}\bar{F}_{0}(x_{1})]^{\theta_{1}}[\bar{F}_{0}(x_{2})]^{\theta_{2}+\theta_{3}% };&x_{1}\leqslant x_{2},\end{array}\right.$ (5)

where $\bar{F}_{0}(x)=P(X>x)$ is the baseline distribution with $\bar{F}_{0}(x)=0$ for $x\leqslant 0$ without loss of generality and $\theta_{1},\theta_{2},\theta_{3}>0$ . In this paper, the functional equation characterized by the singular family of bivariate distributions in Eq. (5) is generalized. The solution of this equation is a very general class of singular family of bivariate distributions of which Eq. (5) is a member. The rest of the paper is organized as below.

Section 2 proposes a class of distributions as a general solution to the functional equation characterizing Eq. (5). This functional equation characterizes a very general class of distributions, including those in Eqs (1)–(5). It is observed through a counter-example that the general solutions of this functional equation need not necessarily generate a bivariate probability distribution function. However, imposing certain conditions on the marginals ensures the solution is a bivariate probability distribution function. Accordingly, in Section 3, necessary and sufficient conditions are derived for univariate probability distributions to qualify as marginals for the bivariate probability distribution belonging to the proposed class. In Section 4, we translate these in terms of failure rates for the ease of constructing bivariate probability distributions satisfying the functional equation. Section 5 discusses the proposed class of probability distributions and its application in simulating a bivariate pair of random variables from a member distribution of the proposed class.

2. Functional equation

In this section, we propose a characterization of Eq. (5) in terms of a functional equation, the general solution of which contains many bivariate singular distributions including Eq. (5).

.

For some baseline survival function $\bar{F}_{0}(\cdot)$ , the functional equation

$\displaystyle\bar{F}(\bar{F}_{0}^{-1}(\bar{F}_{0}(t)\bar{F}_{0}(x_{1})),\bar{F% }_{0}^{-1}(\bar{F}_{0}(t)\bar{F}_{0}(x_{2})))=\bar{F}(x_{1},x_{2})\bar{F}(t,t)$ (6)

is satisfied if and only if

$\displaystyle\bar{F}(x_{1},x_{2})=\left\{\begin{array}[]{ll}\bar{F_{1}}\left(% \bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{1})}{\bar{F}_{0}(x_{2})}\right)% \right)[\bar{F}_{0}(x_{2})]^{\theta};&x_{1}\geqslant x_{2}\\ \\ \bar{F_{2}}\left(\bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{2})}{\bar{F}_{0}(x% _{1})}\right)\right)[\bar{F}_{0}(x_{1})]^{\theta};&x_{1}\leqslant x_{2},\end{% array}\right.$ (7)

for some $\theta>0$ and $x_{1},x_{2},t\geqslant 0$ where $\bar{F_{i}}(x_{i})=\bar{F}(x_{1},x_{2})_{|x_{3-i}=0};i=1,2$ .

Proof..

In Eq. (6), for $x_{1}=x_{2}=x$ , we get

$\displaystyle\bar{F}(\bar{F}_{0}^{-1}(\bar{F}_{0}(t)\bar{F}_{0}(x)),\bar{F}_{0% }^{-1}(\bar{F}_{0}(t)\bar{F}_{0}(x)))=\bar{F}(x,x)\bar{F}(t,t).$ (8)

Writing $\bar{F}(x,x)=H(x)$ , Eq. (8) becomes,

$\displaystyle H(\bar{F}_{0}^{-1}(\bar{F}_{0}(t)\bar{F}_{0}(x)))=H(x)H(t),$ (9)

which is of the form

$\displaystyle T(us)=T(s)T(u),$ (10)

where $s=\bar{F}_{0}(x)$ , $u=\bar{F}_{0}(t)$ and $H\circ\bar{F}_{0}^{-1}=T$ . Writing $u=e^{-a}$ , $s=e^{-b}$ and $T(e^{-a})=G(a)$ for some $a,b\geqslant 0$ , Eq. (10) can be expressed equivalently as

$\displaystyle G(a+b)=G(a)G(b).$ (11)

Note that $0<G(a)<1$ . Since Eq. (11) is satisfied for all $a,b\geqslant 0$ , we can show from Aczél (1966) (page 38) that every solution is either everywhere or nowhere zero. We can now develop the proof in line with the results in Aczél (1966) (page 38). This is achieved by noting that $G(a)$ is a non-negative continuous function. Hence taking logarithms on both sides of Eq. (11) reduces it to the Cauchy functional equation,

$\displaystyle G_{1}(a+b)=G_{1}(a)+G_{1}(b)\text{ for all }a,b\geqslant 0,$ (12)

where $G_{1}(a)=\ln G(a)$ . By induction, it follows directly that, for some positive integer $n$ ,

$\displaystyle G_{1}(na)=nG_{1}(a).$

For any two positive integers $m$ and $n$ , if $a=\frac{m}{n}y$ , then

$\displaystyle G_{1}\left(\frac{m}{n}y\right)=\frac{m}{n}G_{1}(y)\text{ for % some real }y.$ (13)

In particular, for $y=1$ and $G_{1}(1)=\beta$ , where $\beta<0$ , it follows that

$\displaystyle G_{1}(a)=\beta a,$ (14)

for every positive rational $a$ . For $a=0$ , $G_{1}(0)=0$ follows from Eq. (12). Moreover since $G_{1}$ is continuous, taking limits on both sides of Eq. (14) implies $G_{1}(a)=\beta a$ for all non-negative $a$ . This gives $T(u)=u^{-\beta}$ implying

$\displaystyle H(t)=\bar{F}(t,t)=[\bar{F}_{0}(t)]^{\theta}\text{ for all }t>0,$ (15)

where $\theta=-\beta>0$ . Therefore, Eq. (6) can be written as,

$\displaystyle\bar{F}(\bar{F}_{0}^{-1}(\bar{F}_{0}(t)\bar{F}_{0}(x_{1})),\bar{F% }_{0}^{-1}(\bar{F}_{0}(t)\bar{F}_{0}(x_{2})))=\bar{F}(x_{1},x_{2})[\bar{F}_{0}% (t)]^{\theta}.$

For $x_{1}\geqslant x_{2}$ ,

$\displaystyle\bar{F}(x_{1},x_{2})=\bar{F}\left(\bar{F}_{0}^{-1}\left(\bar{F}_{% 0}(x_{2})\frac{\bar{F}_{0}(x_{1})}{\bar{F}_{0}(x_{2})}\right),\bar{F}_{0}^{-1}% (\bar{F}_{0}(x_{2}))\right)=\bar{F}_{1}\left(\bar{F}_{0}^{-1}\left(\frac{\bar{% F}_{0}(x_{1})}{\bar{F}_{0}(x_{2})}\right)\right)[\bar{F}_{0}(x_{2})]^{\theta}.$

Arguing similarly for $x_{1}\leqslant x_{2}$ , Eq. (7) can be retrieved. The converse is direct. ∎

.

Let $X=\text{min}\{X_{1},X_{2}\}$ and $Z_{i}=\bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(X_{i})}{\bar{F}_{0}(X_{3-i})}% \right);i=1,2$ , then the univariate random variables $X$ and $Z_{i}|X_{3-i}<X_{i}$ are independent and the joint function $\bar{F}(x_{1},x_{2})$ in Eq. (7) can be written as the product of the survival functions of $X$ and $Z_{i}|X_{3-i}<X_{i}$ for $i=1,2$ .

Proof..

Writing $X=\text{min}\{X_{1},X_{2}\}$ , the survival function of $X$ follows from Eq. (15) as,

$\displaystyle\bar{F}_{X}(x)=\bar{F}(x,x)=[\bar{F}_{0}(x)]^{\theta};\theta>0.$

Also, writing $Z_{i}=\bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(X_{i})}{\bar{F}_{0}(X_{3-i})}% \right);i=1,2$ , the survival function of $Z_{i}|X_{3-i}<X_{i}$ is given by,

$\displaystyle\bar{F}_{Z_{i}|X_{3-i}<X_{i}}\left(\bar{F}_{0}^{-1}\left(\frac{% \bar{F}_{0}(x_{i})}{\bar{F}_{0}(x_{3-i})}\right)\right)=\bar{F}_{i}\left(\bar{% F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{i})}{\bar{F}_{0}(x_{3-i})}\right)\right)% ;i=1,2.$

The proof then follows from Eq. (7). ∎

The application of this corollary is discussed in detail in Section 5. The next question of interest is whether the general equation $\bar{F}(x_{1},x_{2})$ in Eq. (7) represents a bivariate probability distribution function. The following counter-example shows that it does not always represent a bivariate probability distribution function for any arbitrary marginal distribution.

.

The LFR-exponential class: For the linear failure rate (LFR) marginals specified by,

$\displaystyle\bar{F_{i}}(x_{i})=e^{-(x_{i}+\alpha x_{i}^{2})};x_{i}>0,\alpha>0% ,i=1,2$

and exponential baseline specified by,

$\displaystyle\bar{F}_{0}(x)=e^{-x};x>0,$

the Eq. (6) reduces to

$\displaystyle\bar{F}(x_{1}+t,x_{2}+t)=\bar{F}(x_{1},x_{2})\bar{F}(t,t)$ (16)

with the corresponding solution Eq. (7) as

$\displaystyle\bar{F}(x_{1},x_{2})=\left\{\begin{array}[]{ll}e^{-(x_{1}-x_{2})-% \alpha(x_{1}-x_{2})^{2}-\theta x_{2}};&x_{1}\geqslant x_{2}\\ e^{-(x_{2}-x_{1})-\alpha(x_{2}-x_{1})^{2}-\theta x_{1}};&x_{1}\leqslant x_{2}.% \end{array}\right.$ (17)

On closely examining $\bar{F}(x_{1},x_{2})$ , it can be shown that the rectangular positivity condition for bivariate distributions is violated. For example, $P(1<X_{1}<2,3<X_{2}<5)\ngeq 0$ for the choice of $\alpha=1.5$ and $\theta=3$ , disqualifying it as a bivariate probability survival function. The functional equation Eq. (16) has been well-studied under this perspective in Kulkarni (2006). Hence not all solutions with arbitrary $\bar{F_{i}}(\cdot)$ and $\bar{F}_{0}(\cdot)$ give bivariate probability distribution. Imposing some restrictions on the marginals ensures this. Motivated by this, we develop the conditions to be satisfied by the marginals so that $\bar{F}(x_{1},x_{2})$ is a bivariate survival function.

3. Necessary and sufficient conditions for generating bivariate distributions

In this section, the conditions on univariate distributions to qualify as marginal in Eq. (7) so that Eq. (7) is a bivariate survival function are derived. Let $\Theta=(\theta,{\bm{\alpha}},{\bm{\beta}})$ , where ${\bm{\alpha}}=(\alpha_{1},\alpha_{2},\dots,\alpha_{k})$ is the vector of parameters involved in the baseline $\bar{F}_{0}(\cdot)$ and ${\bm{\beta}}=(\beta_{1},\beta_{2},\dots,\beta_{m})$ is the vector of parameters involved in the marginal $\bar{F_{i}}(\cdot)$ .

.

Let $F_{i}(x)$ be a distribution function with absolutely continuous density $f_{i}(x)$ for which $\lim_{x\rightarrow\infty}f_{i}(x)=0;i=1,2$ . The necessary and sufficient conditions for $\bar{F}(x_{1},x_{2})$ in Eq. (7) be a bivariate probability distribution are that

(i)
$\theta\leqslant u_{1}(\Theta)+u_{2}(\Theta)\leqslant 2\theta$
(ii)
$\frac{\partial}{\partial x_{3-i}}\ln\left(-\frac{\partial}{\partial x_{i}}\bar% {F_{i}}\left(\bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{i})}{\bar{F}_{0}(x_{3-% i})}\right)\right)\right)\leqslant\theta r_{0}(x_{3-i});i=1,2,$

where $u_{i}(\Theta)=\left[f_{i}\left(\bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{i})}% {\bar{F}_{0}(x_{3-i})}\right)\right)\left|\frac{\partial\bar{F}_{0}^{-1}\left(% \frac{\bar{F}_{0}(x_{i})}{\bar{F}_{0}(x_{3-i})}\right)}{\partial(-\ln\bar{F}_{% 0}(x_{3-i}))}\right|\right]_{|x_{i}=x_{3-i}};i=1,2$ .

Proof..

$\bar{F}(x_{1},x_{2})$ will be a bivariate survival function if and only if it can be written as a convex mixture of $\bar{F}_{a}(x_{1},x_{2})$ and $\bar{F}_{s}(x_{1},x_{2})$ where $\bar{F}_{a}(x_{1},x_{2})$ is the absolutely continuous part, $\bar{F}_{s}(x_{1},x_{2})$ is the singular part, and $\bar{F}_{a}(x_{1},x_{2})$ and $\bar{F}_{s}(x_{1},x_{2})$ are survival functions. Observe, for $0\leqslant\alpha\leqslant 1$ ,

$\displaystyle\frac{\partial^{2}\bar{F}(x_{1},x_{2})}{\partial x_{1}\partial x_% {2}}=\alpha f_{a}(x_{1},x_{2})=\left\{\begin{array}[]{ll}{[}\bar{F}_{0}(x_{2})% ]^{\theta}\left[\frac{\partial^{2}\bar{F}_{1}\left(\bar{F}_{0}^{-1}\left(\frac% {\bar{F}_{0}(x_{1})}{\bar{F}_{0}(x_{2})}\right)\right)}{\partial x_{1}\partial x% _{2}}-\theta\frac{f_{0}(x_{2})}{\bar{F}_{0}(x_{2})}\frac{\partial\bar{F}_{1}% \left(\bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{1})}{\bar{F}_{0}(x_{2})}% \right)\right)}{\partial x_{1}}\right];&x_{1}\geqslant x_{2}\\ {[}\bar{F}_{0}(x_{1})]^{\theta}\left[\frac{\partial^{2}\bar{F}_{2}\left(\bar{F% }_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{2})}{\bar{F}_{0}(x_{1})}\right)\right)}{% \partial x_{1}\partial x_{2}}-\theta\frac{f_{0}(x_{1})}{\bar{F}_{0}(x_{1})}% \frac{\partial\bar{F}_{2}\left(\bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{2})}% {\bar{F}_{0}(x_{1})}\right)\right)}{\partial x_{2}}\right];&x_{1}\leqslant x_{% 2}.\end{array}\right.$ (18)

Also,

$\displaystyle\int_{x_{1}\geqslant x_{2}}\alpha f_{a}(x_{1},x_{2})=1-\frac{1}{% \theta}\left[f_{1}\left(\bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{1})}{\bar{F% }_{0}(x_{2})}\right)\right)\left|\frac{\partial\bar{F}_{0}^{-1}\left(\frac{% \bar{F}_{0}(x_{1})}{\bar{F}_{0}(x_{2})}\right)}{\partial(-\ln\bar{F}_{0}(x_{2}% ))}\right|\right]_{|x_{1}=x_{2}}$

and

$\displaystyle\int_{x_{1}\leqslant x_{2}}\alpha f_{a}(x_{1},x_{2})=1-\frac{1}{% \theta}\left[f_{2}\left(\bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{2})}{\bar{F% }_{0}(x_{1})}\right)\right)\left|\frac{\partial\bar{F}_{0}^{-1}\left(\frac{% \bar{F}_{0}(x_{1})}{\bar{F}_{0}(x_{2})}\right)}{\partial(-\ln\bar{F}_{0}(x_{2}% ))}\right|\right]_{|x_{2}=x_{1}}.$

Writing,

$\displaystyle u_{i}(\Theta)=\left[f_{i}\left(\bar{F}_{0}^{-1}\left(\frac{\bar{% F}_{0}(x_{i})}{\bar{F}_{0}(x_{3-i})}\right)\right)\left|\frac{\partial\bar{F}_% {0}^{-1}\left(\frac{\bar{F}_{0}(x_{i})}{\bar{F}_{0}(x_{3-i})}\right)}{\partial% (-\ln\bar{F}_{0}(x_{3-i}))}\right|\right]_{|x_{i}=x_{3-i}};i=1,2$

we have,

$\displaystyle\int_{x_{1}\geqslant x_{2}}\alpha f_{a}(x_{1},x_{2})=1-\frac{1}{% \theta}u_{1}(\Theta)$

and

$\displaystyle\int_{x_{1}\leqslant x_{2}}\alpha f_{a}(x_{1},x_{2})=1-\frac{1}{% \theta}u_{2}(\Theta)$

so that

$\displaystyle\alpha=2-\frac{1}{\theta}(u_{1}(\Theta)+u_{2}(\Theta)).$ (19)

Hence the absolutely continuous part has density $f_{a}(x_{1},x_{2})$ given by Eqs (18) and (19). Also,

$\displaystyle\bar{F}_{a}(x,x)=[\bar{F}_{0}(x)]^{\theta}.$

But $\bar{F}(x,x)=[\bar{F}_{0}(x)]^{\theta}$ , so that $\bar{F}_{s}(x,x)=\frac{\bar{F}(x,x)-\alpha\bar{F}_{a}(x,x)}{1-\alpha}=[\bar{F}% _{0}(x)]^{\theta}$ . Therefore, $\bar{F}$ is a valid survival function if

(i)
$\bar{F}$ is a convex mixture of $F_{a}$ and $F_{s}$ . From Eq. (19), since $0\leqslant\alpha\leqslant 1$ , it follows that

$\displaystyle\theta\leqslant u_{1}(\Theta)+u_{2}(\Theta)\leqslant 2\theta.$
(ii)
$\bar{F}_{a}$ is a valid survival function. From Eq. (18), since $f_{a}(x_{1},x_{2})\geqslant 0$ , for $x_{i}>x_{3-i};i=1,2$ , it follows that

$\displaystyle\frac{\partial}{\partial x_{3-i}}\ln\left(-\frac{\partial}{% \partial x_{i}}\bar{F_{i}}\left(\bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{i})% }{\bar{F}_{0}(x_{3-i})}\right)\right)\right)\leqslant\theta r_{0}(x_{3-i}).$

This completes the proof. ∎

Counter-Example 2.1 (contd.) The LFR-exponential class: Consider again the bivariate function $\bar{F}(x_{1},x_{2})$ given in Eq. (17). Condition (ii) of Theorem 3.1 becomes

$\displaystyle 2\alpha(x_{i}-x_{3-i})-\frac{1}{x_{i}-x_{3-i}}+1\leqslant\theta;% x_{i}>x_{3-i},i=1,2,$

which is not satisfied for the choice of $\alpha=1.5,\theta=3,x_{1}=5$ and $x_{2}=3$ . This again shows that a bivariate probability distribution cannot be formed with marginal being the linear failure rate model and baseline as exponential distribution in Eq. (7). However, any proportional hazard distribution qualifies as marginals in Eq. (7). This is discussed in detail in the next theorem.

.

The survival function $\bar{F}(x_{1},x_{2})$ with the marginals belonging to the PH class satisfies the functional equation Eq. (6) if and only if it is of the form

$\displaystyle\bar{F}(x_{1},x_{2})=\left\{\begin{array}[]{ll}[\bar{F}_{0}(x_{1}% )]^{\delta_{1}}[\bar{F}_{0}(x_{2})]^{\theta-\delta_{1}};&x_{1}\geqslant x_{2}% \\ {[}\bar{F}_{0}(x_{1})]^{\theta-\delta_{2}}[\bar{F}_{0}(x_{2})]^{\delta_{2}};&x% _{1}\leqslant x_{2},\end{array}\right.$ (20)

for some $\theta,\delta_{1},\delta_{2}>0$ with $\delta_{i}<\theta;i=1,2$ .

Proof..

Let Eq. (6) be satisfied. Then, the general solution of $\bar{F}(x_{1},x_{2})$ is given by Eq. (7). If the marginals belong to the PH class so that $\bar{F_{i}}(x_{i})=[\bar{F}_{0}(x_{i})]^{\delta_{i}};\delta_{i}>0,i=1,2$ , then the bivariate function in Eq. (7) reduces to Eq. (20). Now, it remains to show that $\bar{F}(x_{1},x_{2})$ is a valid bivariate probability distribution. Proceeding as in Theorem 3.1,

$\displaystyle\frac{\partial^{2}\bar{F}(x_{1},x_{2})}{\partial x_{1}\partial x_% {2}}=\alpha f_{a}(x_{1},x_{2})=\left\{\begin{array}[]{ll}\delta_{1}(\theta-% \delta_{1})f_{0}(x_{1})f_{0}(x_{2})(\bar{F}_{0}(x_{1}))^{\delta_{1}-1}(\bar{F}% _{0}(x_{2}))^{\theta-\delta_{1}-1};&x_{1}\geqslant x_{2}\\ \delta_{2}(\theta-\delta_{2})f_{0}(x_{1})f_{0}(x_{2})(\bar{F}_{0}(x_{1}))^{% \theta-\delta_{2}-1}(\bar{F}_{0}(x_{2}))^{\delta_{2}-1};&x_{1}\leqslant x_{2}.% \end{array}\right.$ (21)

Also,

$\displaystyle\int_{x_{1}\geqslant x_{2}}\alpha f_{a}(x_{1},x_{2})=\frac{\theta% -\delta_{1}}{\theta}$

and

$\displaystyle\int_{x_{1}\leqslant x_{2}}\alpha f_{a}(x_{1},x_{2})=\frac{\theta% -\delta_{2}}{\theta}$

so that $\alpha=\frac{2\theta-\delta_{1}-\delta_{2}}{\theta}$ . Hence, the absolute continuous part has density given by,

$\displaystyle f_{a}(x_{1},x_{2})=\frac{\theta}{2\theta-\delta_{1}-\delta_{2}}% \times\left\{\begin{array}[]{ll}\delta_{1}(\theta-\delta_{1})f_{0}(x_{1})f_{0}% (x_{2})(\bar{F}_{0}(x_{1}))^{\delta_{1}-1}(\bar{F}_{0}(x_{2}))^{\theta-\delta_% {1}-1};&x_{1}\geqslant x_{2}\\ \delta_{2}(\theta-\delta_{2})f_{0}(x_{1})f_{0}(x_{2})(\bar{F}_{0}(x_{1}))^{% \theta-\delta_{2}-1}(\bar{F}_{0}(x_{2}))^{\delta_{2}-1};&x_{1}\leqslant x_{2}.% \end{array}\right.$ (22)

Note that $\bar{F}_{a}(x_{1},x_{2})=[\bar{F}_{0}(x)]^{\theta}$ . But, $\bar{F}(x,x)=[\bar{F}_{0}(x)]^{\theta}$ so that $\bar{F}_{s}(x_{1},x_{2})=\frac{\bar{F}(x,x)-\alpha\bar{F}_{a}(x,x)}{1-\alpha}=% [\bar{F}_{0}(x)]^{\theta}$ . From (i) and (ii), as in Theorem 3.1, we have $\theta\leqslant\delta_{1}+\delta_{2}\leqslant 2\theta$ , which is true since $\delta_{i}<\theta;∼{}i=1,2$ . Also, $\bar{F}_{a}$ is a valid survival function implying $f_{a}(x_{1},x_{2})\geqslant 0$ , which is true for any choice of the baseline distribution $\bar{F}_{0}(\cdot)$ . The converse is direct. ∎
4. Construction of bivariate probability distributions satisfying functional equation Eq. (6)

The functional equation Eq. (6) can be equivalently represented in terms of the hazard gradient. The discussions above translated in terms of the hazard gradient would make it computationally easier to check if marginals qualify for the function Eq. (7) to be a bivariate survival function. Accordingly, in this section, the functional equation in terms of its hazard gradient is studied. The hazard gradient vector $r(x_{1},x_{2})$ (Johnson & Kotz, 1975) is defined as

$\displaystyle r(x_{1},x_{2})=(r_{1}(x_{1},x_{2}),r_{2}(x_{1},x_{2}))=\left(-% \frac{\partial\ln\bar{F}(x_{1},x_{2})}{\partial x_{1}},-\frac{\partial\ln\bar{% F}(x_{1},x_{2})}{\partial x_{2}}\right)=\left(\frac{\partial}{\partial x_{1}}R% (x_{1},x_{2}),\frac{\partial}{\partial x_{2}}R(x_{1},x_{2})\right),$

where $R(x_{1},x_{2})=-\ln\bar{F}(x_{1},x_{2})$ .

Note that the functional equation Eq. (6) can be equivalently expressed as

$\displaystyle R(\bar{F}_{0}^{-1}(\bar{F}_{0}(t)\bar{F}_{0}(x_{1})),\bar{F}_{0}% ^{-1}(\bar{F}_{0}(t)\bar{F}_{0}(x_{2})))=R(x_{1},x_{2})-\theta\ln\bar{F}_{0}(t).$

Differentiate with respect to $t$ , to get,

$\displaystyle\sum_{i=1}^{2}\frac{\partial\big{[}R(\bar{F}_{0}^{-1}(\bar{F}_{0}% (t)\bar{F}_{0}(x_{1})),\bar{F}_{0}^{-1}(\bar{F}_{0}(t)\bar{F}_{0}(x_{2})))]}{% \partial(\bar{F}_{0}^{-1}(\bar{F}_{0}(t)\bar{F}_{0}(x_{i})))}\frac{\partial(% \bar{F}_{0}^{-1}(\bar{F}_{0}(t)\bar{F}_{0}(x_{i})))}{\partial t}=\theta r_{0}(t)$

which gives a condition in terms of the hazard gradient to satisfy the functional equation Eq. (6) as

$\displaystyle\sum_{i=1}^{2}r_{i}(\bar{F}_{0}^{-1}(\bar{F}_{0}(t)\bar{F}_{0}(x_% {1})),\bar{F}_{0}^{-1}(\bar{F}_{0}(t)\bar{F}_{0}(x_{2})))\frac{\partial(\bar{F% }_{0}^{-1}(\bar{F}_{0}(t)\bar{F}_{0}(x_{i})))}{\partial t}=\theta r_{0}(t),$ (23)

where $r_{0}(\cdot)$ is the baseline hazard function. On integrating Eq. (23), the functional equation Eq. (6) is retrieved.

.

A bivariate random vector $X=(X_{1},X_{2})$ satisfy Eq. (6) if and only if

$\displaystyle r_{i}(x_{1},x_{2})=\left\{\begin{array}[]{ll}{r_{i}\left(\bar{F}% _{0}^{-1}\left(\frac{\bar{F}_{0}(x_{i})}{\bar{F}_{0}(x_{3-i})}\right)\right)% \frac{\partial}{\partial x_{i}}\left(\bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x% _{i})}{\bar{F}_{0}(x_{3-i})}\right)\right)};&x_{i}\geqslant x_{3-i}\\ {-r_{3-i}\left(\bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{3-i})}{\bar{F}_{0}(x% _{i})}\right)\right)\left|\frac{\partial}{\partial x_{i}}\left(\bar{F}_{0}^{-1% }\left(\frac{\bar{F}_{0}(x_{3-i})}{\bar{F}_{0}(x_{i})}\right)\right)\right|+% \theta r_{0}(x_{i})};&x_{i}\leqslant x_{3-i},\end{array}\right.$ (24)

where $r_{i}(x)$ is the marginal hazard rate of $X_{i};i=1,2$ .

Proof..

Suppose $(X_{1},X_{2})$ satisfy,

$\displaystyle\bar{F}(\bar{F}_{0}^{-1}(\bar{F}_{0}(t)\bar{F}_{0}(x_{1})),\bar{F% }_{0}^{-1}(\bar{F}_{0}(t)\bar{F}_{0}(x_{2})))=\bar{F}(x_{1},x_{2})[\bar{F}_{0}% (t)]^{\theta}.$

Then,

$\displaystyle\bar{F}(x_{1},x_{2})=\bar{F}\left(\bar{F}_{0}^{-1}\left(\frac{% \bar{F}_{0}(x_{1})}{\bar{F}_{0}(t)}\right),\bar{F}_{0}^{-1}\left(\frac{\bar{F}% _{0}(x_{2})}{\bar{F}_{0}(t)}\right)\right)[\bar{F}_{0}(t)]^{\theta}.$

For $t=\text{min}\{x_{1},x_{2}\}$ and $x_{1}\geqslant x_{2}$ , it follows that,

$\displaystyle\bar{F}(x_{1},x_{2})=\bar{F}\left(\bar{F}_{0}^{-1}\left(\frac{% \bar{F}_{0}(x_{1})}{\bar{F}_{0}(x_{2})}\right),0\right)[\bar{F}_{0}(x_{2})]^{% \theta}=\bar{F}_{1}\left(\bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{1})}{\bar{% F}_{0}(x_{2})}\right)\right)[\bar{F}_{0}(x_{2})]^{\theta}.$

On differentiating w.r.t $x_{i};i=1,2$ , it further follows that,

$\displaystyle r_{1}(x_{1},x_{2})=r_{1}\left(\bar{F}_{0}^{-1}\left(\frac{\bar{F% }_{0}(x_{1})}{\bar{F}_{0}(x_{2})}\right)\right)\frac{\partial}{\partial x_{1}}% \left(\bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{1})}{\bar{F}_{0}(x_{2})}% \right)\right)$

and

$\displaystyle r_{2}(x_{1},x_{2})=-r_{1}\left(\bar{F}_{0}^{-1}\left(\frac{\bar{% F}_{0}(x_{1})}{\bar{F}_{0}(x_{2})}\right)\right)\left|\frac{\partial}{\partial x% _{2}}\left(\bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{1})}{\bar{F}_{0}(x_{2})}% \right)\right)\right|+\theta r_{0}(x_{2}).$

Proceed in a similar manner for $x_{1}\leqslant x_{2}$ to obtain Eq. (24). The converse is straight forward through the representation,

$\displaystyle\bar{F}(x_{1},x_{2})=\exp\left[-\int_{0}^{x_{1}}r_{1}(u,0)du-\int% _{0}^{x_{2}}r_{2}(x_{1},u)du\right]$

$\displaystyle\bar{F}(x_{1},x_{2})=\exp\left[-\int_{0}^{x_{1}}r_{1}(u,x_{2})du-% \int_{0}^{x_{2}}r_{2}(0,u)du\right]$

(Johnson & Kotz, 1975). ∎

On close investigation of Eq. (24), observe that the marginal hazard function $r_{i}(x)$ should satisfy the condition,

$\displaystyle r_{i}\left(\bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{i})}{\bar{% F}_{0}(x_{3-i})}\right)\right)\left|\frac{\partial}{\partial x_{3-i}}\left(% \bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{i})}{\bar{F}_{0}(x_{3-i})}\right)% \right)\right|\leqslant\theta r_{0}(x_{3-i});\text{ for }i=1,2$ (25)

since $r_{i}(x_{1},x_{2})\geqslant 0$ . Based on these observations, the necessary and sufficient conditions for differentiable $r_{i}(x)$ to qualify as marginal failure rates for $\bar{F}(x_{1},x_{2})$ to be a bivariate survival function are discussed in the following theorem.

.

The necessary and sufficient conditions for differentiable functions $r_{i}(x);i=1,2$ to qualify as marginal failure rates of $\bar{F}(x_{1},x_{2})$ in Eq. (7) are that for $x_{i}>x_{3-i};i=1,2$ ,

(i)

$0\leqslant r_{i}\left(\bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{i})}{\bar{F}_% {0}(x_{3-i})}\right)\right)\left|\frac{\partial\bar{F}_{0}^{-1}\left(\frac{% \bar{F}_{0}(x_{i})}{\bar{F}_{0}(x_{3-i})}\right)}{\partial x_{3-i}}\right|% \leqslant\theta r_{0}(x_{3-i})$

(ii)

$\int_{0}^{\infty}r_{i}(x)dx=\infty$

(iii)

$r_{i}\left(\bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{i})}{\bar{F}_{0}(x_{3-i}% )}\right)\right)\frac{\partial\bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{i})}{% \bar{F}_{0}(x_{3-i})}\right)}{\partial x_{i}}\left[\theta r_{0}(x_{3-i})+r_{i}% \left(\bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{i})}{\bar{F}_{0}(x_{3-i})}% \right)\right)\frac{\partial\bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{i})}{% \bar{F}_{0}(x_{3-i})}\right)}{\partial x_{3-i}}\right]-$

$\frac{\partial}{\partial x_{3-i}}\left[r_{i}\left(\bar{F}_{0}^{-1}\left(\frac{% \bar{F}_{0}(x_{i})}{\bar{F}_{0}(x_{3-i})}\right)\right)\frac{\partial\bar{F}_{% 0}^{-1}\left(\frac{\bar{F}_{0}(x_{i})}{\bar{F}_{0}(x_{3-i})}\right)}{\partial x% _{i}}\right]\geqslant 0$

(iv)

$\theta\leqslant v_{1}(\Theta)+v_{2}(\Theta)\leqslant 2\theta$

where $v_{i}(\Theta)=\left[r_{i}\left(\bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{i})}% {\bar{F}_{0}(x_{3-i})}\right)\right)\exp\left\{-\int_{0}^{\bar{F}_{0}^{-1}% \left(\frac{\bar{F}_{0}(x_{i})}{\bar{F}_{0}(x_{3-i})}\right)}r_{i}(u)du\right% \}\left|\frac{\partial\bar{F}_{0}^{-1}\left(\frac{\bar{F}_{0}(x_{i})}{\bar{F}_% {0}(x_{3-i})}\right)}{\partial(-\ln\bar{F}_{0}(x_{3-i}))}\right|\right]_{|x_{i% }=x_{3-i}};i=1,2$ . The bivariate function $\bar{F}(x_{1},x_{2})$ given by Eq. (7) is a survival function satisfying the functional equation in Eq. (6) where

$\displaystyle\bar{F}_{i}(x)=\exp\left(-\int_{0}^{x}r_{i}(u)du\right);x% \geqslant 0,i=1,2.$ (26)

Proof..

Condition (i) follows from Eq. (25), and condition (ii) is necessary for $r_{i};i=1,2$ to be a hazard rate function. Conditions (iii) and (iv) imply conditions (i) and (ii) of Theorem 3.1.

Conversely, the hazard gradient associated with $\bar{F}(x_{1},x_{2})$ is given by Eq. (24), and condition (i) follows from the non-negativity of $r_{i}(x_{1},x_{2});i=1,2$ . Since $r_{i}(\cdot);i=1,2$ are univariate failure rates, condition (ii) follows. Condition (iii) follows from the non-negativity of the density function associated with $\bar{F}(x_{1},x_{2})$ , and condition (iv) follows from Theorem 3.1. ∎

Counter-Example 2.1 (contd.) The LFR-exponential class: The marginal failure rates are given by

$\displaystyle r_{i}(x_{i})=2\alpha x_{i}+1;i=1,2.$

Then, condition (i) in Theorem 4.2 becomes $0\leqslant 2\alpha(x_{i}-x_{3-i})+1\leqslant\theta;x_{i}>x_{3-i}$ , which is not satisfied for the choice of $\alpha=1.5,\theta=3,x_{1}=5$ and $x_{2}=3$ . Hence, we cannot construct a bivariate survival function $\bar{F}(x_{1},x_{2})$ with the marginals as linear failure rate and baseline distribution as exponential.

.

The BPHC-Weibull class: For a proportional hazard rate,

$\displaystyle r_{i}(x_{i})=(\theta_{i}+\theta_{3})r_{0}(x_{i});x_{i}>0,i=1,2$

with $\theta_{i}>0;i=1,2,3$ and $r_{0}(x)=\alpha x^{\alpha-1};x>0,\alpha>0$ , it is easily seen that conditions of Theorem 4.2 are satisfied. For,

(i)

$0\leqslant\theta_{i}+\theta_{3}\leqslant\theta_{1}+\theta_{2}+\theta_{3}$

(ii)

$\int_{0}^{\infty}\alpha(\theta_{i}+\theta_{3})x_{i}^{\alpha-1}dx_{i}=\infty$

(iii)

$\theta_{i}+\theta_{3}\geqslant 0$

(iv)

$\theta_{1}+\theta_{2}+\theta_{3}\leqslant\theta_{1}+\theta_{2}+2\theta_{3}% \leqslant 2(\theta_{1}+\theta_{2}+\theta_{3})$ .

for $x_{i}>x_{3-i};i=1,2$ . From Eq. (26), $\bar{F}_{i}(x_{i})=[\bar{F}_{0}(x_{i})]^{\theta_{i}+\theta_{3}};i=1,2$ , where $\bar{F}_{0}(x)=e^{-x^{\alpha}};x>0,\alpha>0$ . Substituting in Eq. (7), we get

5. Discussion

A class of distributions has been proposed as a general solution to the functional equation Eq. (6). This class of distribution is specified by,

$\displaystyle\bar{F}(x_{1},x_{2})=\left\{\begin{array}[]{ll}\bar{F_{1}}\Big{(}% \bar{F}_{0}^{-1}\big{(}\frac{\bar{F}_{0}(x_{1})}{\bar{F}_{0}(x_{2})}\big{)}% \Big{)}[\bar{F}_{0}(x_{2})]^{\theta};&x_{1}\geqslant x_{2}\\ \\ \bar{F_{2}}\Big{(}\bar{F}_{0}^{-1}\big{(}\frac{\bar{F}_{0}(x_{2})}{\bar{F}_{0}% (x_{1})}\big{)}\Big{)}[\bar{F}_{0}(x_{1})]^{\theta};&x_{1}\leqslant x_{2},\end% {array}\right.$

for some $\theta>0$ and $x_{1},x_{2},t\geqslant 0$ , where $\bar{F_{i}}(x_{i})=\bar{F}(x_{1},x_{2})_{|x_{3-i}=0};i=1,2$ and $\bar{F}_{0}(\cdot)$ is the baseline distribution. A counter-example is given, showing that not all marginals admit a bivariate probability distribution in this class. The conditions to be satisfied by a univariate distribution to be a marginal of the bivariate class are developed. These conditions are translated in terms of the univariate failure rates enabling them to be easily used to generate a bivariate probability distribution belonging to the general class. Many well-known classes of distributions are members of the proposed class in Eq. (7). Some of the well-studied classes are given in the following examples.

5.1 Example 1 – Extended Freund’s bivariate distribution (Asha et al., 2016)

If the marginals are specified by

$\displaystyle\bar{F}_{i}(x_{i})=\left\{\begin{array}[]{ll}\frac{\theta_{3-i}}{% \theta_{i}+\theta_{3-i}-\theta^{\prime}_{i}}[\bar{F}_{0}(x_{i})]^{\theta^{% \prime}_{i}}+\frac{\theta_{i}-\theta^{\prime}_{i}}{\theta_{i}+\theta_{3-i}-% \theta^{\prime}_{i}}[\bar{F}_{0}(x_{i})]^{\theta_{i}+\theta_{3-i}};&\theta_{i}% +\theta_{3-i}\neq\theta^{\prime}_{i}\\ {[}\bar{F}_{0}(x_{i})]^{\theta_{1}+\theta_{2}}[1-\theta_{3-i}\ln\bar{F}_{0}(x_% {i})];&\theta_{i}+\theta_{3-i}=\theta^{\prime}_{i}\end{array}\right.$

for $x_{i}>0;i=1,2$ where $\theta_{1},\theta_{2},\theta^{\prime}_{1},\theta^{\prime}_{2}>0$ and rewriting $\theta=\theta_{1}+\theta_{2}$ , the solution in Eq. (7) reduces to

$\displaystyle\bar{F}(x_{1},x_{2})=\left\{\begin{array}[]{ll}[\bar{F}_{0}(x_{i}% )]^{\theta_{i}+\theta_{3-i}}+\frac{\theta_{3-i}[\bar{F}_{0}(x_{i})]^{\theta^{% \prime}_{i}}}{\theta_{i}+\theta_{3-i}-\theta^{\prime}_{i}}[(\bar{F}_{0}(x_{3-i% }))^{\theta_{i}+\theta_{3-i}-\theta^{\prime}_{i}}-(\bar{F}_{0}(x_{i}))^{\theta% _{i}+\theta_{3-i}-\theta^{\prime}_{i}}];&x_{i}>x_{3-i},\\ &\theta_{i}+\theta_{3-i}\neq\theta^{\prime}_{i}\\ {[}\bar{F}_{0}(x_{i})]^{\theta_{1}+\theta_{2}}[1+\theta_{3-i}(\ln\bar{F}_{0}(x% _{3-i})-\ln\bar{F}_{0}(x_{i}))];&x_{i}>x_{3-i},\\ &\theta_{i}+\theta_{3-i}=\theta^{\prime}_{i}.\end{array}\right.$ (27)

5.2 Example 2 – Bivariate semi-parametric singular family of distributions (Kundu, 2022)

If the marginals are specified by

$\displaystyle\bar{F}_{i}(x_{i})=[\bar{F}_{0}(x_{i})]^{\theta_{i}+\theta_{3}};x% _{i}>0,i=1,2$

where $\theta_{1},\theta_{2},\theta_{3}>0$ and rewriting $\theta=\theta_{1}+\theta_{2}+\theta_{3}$ , the solution in Eq. (7) reduces to

$\displaystyle\bar{F}(x_{1},x_{2})=\left\{\begin{array}[]{ll}[\bar{F}_{0}(x_{1}% )]^{\theta_{1}+\theta_{3}}[\bar{F}_{0}(x_{2})]^{\theta_{2}};&x_{1}\geqslant x_% {2}\\ {[}\bar{F}_{0}(x_{1})]^{\theta_{1}}[\bar{F}_{0}(x_{2})]^{\theta_{2}+\theta_{3}% };&x_{1}\leqslant x_{2}.\end{array}\right.$

Apart from this, the distributional properties and inference procedures of several probability distributions belonging to the class in Eq. (7) are discussed in the literature (see Proschan and Sullo (1976), Pena and Gupta (1990), Asha et al. (2016), Kundu (2022) to mention a few). For these studies, it is necessary to generate pairs of random variables from a continuous bivariate distribution. While specific simulation techniques are available for certain bivariate distributions, such as bivariate gamma, exponential, and normal distributions, simulating from a general bivariate distribution can often present multiple challenges. Theorem 2.1 provides a characterization result that aids in simulating a pair of random variables whose distribution satisfies the functional equation Eq. (6) through Corollary 2.1. Here, we can generate a bivariate pair $(X_{1},X_{2})$ by simulating $X$ and $Z$ and using the probability $P(X_{i}<X_{3-i});i=1,2$ for a given bivariate distribution satisfying the functional equation Eq. (6). This method also simplifies testing the goodness of fit of a bivariate data set, as it only requires testing of univariate quantities.

Footnotes

Acknowledgments

The authors thank the referee for the meticulous reading that has tremendously improved the quality of the paper. This work was financially supported by the Council of Scientific & Industrial Research (CSIR), Government of India, through the Senior Research Fellowship scheme vide No 09/239(0551)/2019-EMR-I.

Conflict of interest

The authors have no conflicts of interest to declare.

References

Aczél

(1966). Lectures on functional equations and their applications. Academic press.

Asha

Krishna KM

, & Kundu

(2016). An extension of the Freund’s bivariate distribution to model load-sharing systems. American Journal of Mathematical and Management Sciences, 35(3), 207-226.

Galambos

, & Kotz

(2006). Characterizations of Probability Distributions.: A Unified Approach with an Emphasis on Exponential and Related Models, Vol. 675. Springer.

Johnson

N.L.

, & Kotz

(1975). A vector multivariate hazard rate. Journal of Multivariate Analysis, 5(1), 53-66.

Kolev

, & Pinto

(2018a). Functional equations involving Sibuya’s dependence function. Aequationes Mathematicae, 92(3), 441-451.

Kolev

, & Pinto

(2018b). A weak version of bivariate lack of memory property. Brazilian Journal of Probability and Statistics, 32(4), 873-906.

Kulkarni

(2006). Characterizations and modelling of multivariate lack of memory property. Metrika, 64(2), 167-180.

Kundu

(2022). Bivariate semi-parametric singular family of distributions and its applications. Sankhya B, 1-27.

Lin

G.D.

Dou

, & Kuriki

(2019). The bivariate lack-of-memory distributions. Sankhya A, 81(2), 273-297.

10.

Marshall

A.W.

, & Olkin

(1967). A multivariate exponential distribution. Journal of the American Statistical Association, 62(317), 30-44.

11.

Nelsen

R.B.

(2007). An introduction to copulas. Springer Science & Business Media.

12.

Pena

E.A.

, & Gupta

A.K.

(1990). Bayes estimation for the Marshall-Olkin exponential distribution. Journal of the Royal Statistical Society: Series B (Methodological), 52(2), 379-389.

13.

Pinto

, & Kolev

(2015). Sibuya-type bivariate lack of memory property. Journal of Multivariate Analysis, 134, 119-128.

14.

Proschan

, & Sullo

(1976). Estimating the parameters of a multivariate exponential distribution. Journal of the American Statistical Association, 71(354), 465-472.