Further Results on the Bivariate Semi-parametric Singular Family of Distributions

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

where $0<a,b<1$ and $0<u_{i}<1;\leavevmode\nobreak\ i=1,2$ is apt in modelling association of these types (see Nelsen (2007)). Many authors have proposed very general classes of bivariate distributions and studied their properties. In (1.1), when $u_{i}$ is exponential($\theta_{i}+\theta_{3}$) for $i=1,2$ and $a=\frac{\theta_{3}}{\theta_{1}+\theta_{3}}$ and $b=\frac{\theta_{3}}{\theta_{2}+\theta_{3}}$, the well studied Marshall-Olkin bivariate exponential distribution with a singular component is obtained as

where $\theta_{i}>0;\leavevmode\nobreak\ i=1,2,3$. Here $(X_{1},X_{2})$ is a bivariate random vector with $\bar{F}(x_{1},x_{2})=P(X_{1}>x_{1},X_{2}>x_{2})$ with support $I_{(0,\infty)\times(0,\infty)}=\{(x_{1},x_{2});\leavevmode\nobreak\ 0<x_{1}<\infty,\leavevmode\nobreak\ 0<x_{2}<\infty\}$. The distribution in (1.2) is characterised by the functional equation popularly known as bivariate lack of memory property (BLMP) given by

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

If $u_{i}$ is Weibull($\theta_{i}+\theta_{3},\alpha$) for $i=1,2$, the Marshall-Olkin copula in (1.1) gives a bivariate distribution with a singular component whose survival function is

where $\alpha,\leavevmode\nobreak\ \theta_{i}>0;\leavevmode\nobreak\ i=1,2,3$, characterised by the functional equation,

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

If $u_{i}$ is Pareto($\theta_{i}+\theta_{3}$) for $i=1,2$, the Marshall-Olkin copula in (1.1) gives a bivariate distribution with a singular component whose survival function is

where $\theta_{i}>0;\leavevmode\nobreak\ i=1,2,3$, characterised by the functional equation,

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

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 (1.1) becomes the copula associated with the bivariate proportional hazard class (BPHC) mentioned in Kundu (2022) whose bivariate survival function is

where $\bar{F}_{0}(x)=P(X>x)$ is the baseline distribution with $\bar{F}_{0}(x)=0$ for $x\leq 0$ without loss of generality and $\theta_{i}>0;\leavevmode\nobreak\ i=1,2,3$. In this work, we study the functional equation characterised by singular family of bivariate distributions in (1.5). The rest of the paper is organised as below.

In Section 2, we propose a class of distributions as a general solution to the functional equation characterising (1.5). This functional equation characterises a very general class of distributions which includes (1.2), (1.3), (1.4) and (1.5). It is observed through a counter example that the general solutions of this functional equation need not necessarily generate a bivariate distribution function. But conditions on the marginals ensures that the solution is a bivariate distribution. In Section 3, necessary and sufficient conditions are derived for a univariate distribution to be a marginal for the bivariate distribution belonging to this class. In Section 4, we translate these in terms of failure rates for the ease of constructing bivariate distributions satisfying the functional equation.

In this section, we propose a characterisation of (1.5) in terms of a functional equation, the general solution of which includes many bivariate singular distributions including (1.5).

The next question of interest is whether this general equation $\bar{F}(x_{1},x_{2})$ in (2.2) represent proper bivariate distribution function? The following counter example shows it is not. It is further seen that arbitrary marginals do not admit a proper bivariate distribution.

In this section, we discuss the conditions to be satisfied by the univariate distributions for them to qualify as a marginal for the bivariate survival function as in (2.2). Let $\Theta=(\theta,\mathbold{\alpha},\mathbold{\beta})$, where $\mathbold{\alpha}=(\alpha_{1},\alpha_{2},\dots,\alpha_{k})$ is the vector of parameters involved in baseline $\bar{F}_{0}(\cdot)$ and $\mathbold{\beta}=(\beta_{1},\beta_{2},\dots,\beta_{m})$ is the vector of parameters involved in the marginal $\bar{F_{i}}(\cdot)$.

Counter Example 2.1 (contd.) LFR-exponential: Consider again the bivariate function $\bar{F}(x_{1},x_{2})$ given in (2.6). Condition (ii) of Theorem 3.2 becomes

which is not satisfied for the choice of $\alpha=1.5,\leavevmode\nobreak\ \theta=3,\leavevmode\nobreak\ x_{1}=5$ and $x_{2}=3$. This again shows that a bivariate distribution cannot be formed with marginal being the linear failure rate model and baseline as exponential distribution in (2.2). However any proportional hazard distribution qualify as marginals in (2.2). This is discussed in detail in the next theorem.

The functional equation (2.1) 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 (2.2) to be a bivariate survival function. Accordingly in this section we visit the functional equation in terms of its hazard gradient. The hazard gradient vector $r(x_{1},x_{2})$ (Johnson and Kotz (1975)) is defined as

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

Note that the functional equation (2.1) can be equivalently expressed as

Differentiating with respect to $t$, we get,

which gives a condition in terms of the hazard gradient to satisfy the functional equation (2.1) as

where $r_{0}(\cdot)$ is the baseline hazard function. On integrating (4.1), we can retrieve the functional equation (2.1).