Relation of survival and non-survival Marshall-Olkin copula

Question

Let us have two random variables $A$ and $B$ representing lifetimes of two elements of a system, where $A$ has cdf $F_A(x)$, $A \sim Exp(\lambda_1 + \lambda_{12})$ and $B$ has cdf $F_B(y)$, $B \sim Exp(\lambda_2 + \lambda_{12})$, with joint cdf $H(x,y)$ .

Marshall-Olkin copula is defined as survival copula
$$\bar{H}(x,y) = C_{\theta_A, \theta_B}(u,v) = \min (u^{1-\theta_A}v, uv^{1-\theta_B})$$

where $$\theta_A = \frac{\lambda_{12}}{\lambda_1+\lambda_{12}} \text{ and } \theta_B = \frac{\lambda_{12}}{\lambda_2+\lambda_{12}}$$

$$u = \bar{F_A}(x) = 1 - F_A(x) \text{ and }v = \bar{F_B}(y) = 1- F_B(y)$$

What is the formula for non-survival Marshall-Olkin copula that would have $u = F_A(x)$ and $v=F_B(y)$?

Answer 1

Note that the survival copula $C_{\theta_A, \theta_B}(u, v)$ and the non-survival copula $C(u, v)$ are related by \begin{align*} C_{\theta_A, \theta_B}(\hat{u}, \hat{v}) = \hat{u}+\hat{v}-1 + C(1-\hat{u}, 1-\hat{v}), \end{align*} where $\hat{u}=\bar{F}_A(x)=1 - F_A(x)$ and $\hat{v}=\bar{F}_B(y)=1-F_B(y)$. Then, \begin{align*} C(u, v) = C_{\theta_A, \theta_B}(1-u, 1-v) + u+v-1, \end{align*} where $u=F_A(x)$ and $v=F_B(y)$.