I am trying to define a bivariate exponential distribution $(S, T)$ with marginals $S\sim\mathrm{Exp}(\lambda_S)$ and $T\sim\mathrm{Exp}(\lambda_T)$ for $\lambda_S > \lambda_T$. I would like the joint distribution of $S$ and $T$ to have two properties:
- $\rho(S, T)$ can be controlled by some parameter of the joint distribution
- $S\leq T$
The literature has many examples of distributions that fulfill one of these two properties. For instance, the BVE of Marshall and Olkin (1967) can be constructed from independent random variables $\tilde S, \tilde T$, and $C$ and selected constant $\lambda_C < \lambda_T$ as
\begin{align*} \tilde S &\sim \mathrm{Exp}(\lambda_S-\lambda_C) \\ \tilde T &\sim \mathrm{Exp}(\lambda_T-\lambda_C) \\ C &\sim \mathrm{Exp}(\lambda_C) \\ S &= \min(\tilde S, C) \\ T &= \min(\tilde T, C) \end{align*}
Here, $S$ and $T$ have the desired marginal distributions, and $\lambda_C$ controls the degree to which they are correlated (requirement 1). However, we cannot guarantee that $S\leq T$ (requirement 2).
Alternately, we could use independent random variables $\tilde S$ and $T$ to construct
\begin{align*} \tilde S &\sim \mathrm{Exp}(\lambda_S-\lambda_T) \\ T &\sim \mathrm{Exp}(\lambda_T) \\ S &= \min(\tilde S, T) \end{align*}
Again, $S$ and $T$ have the desired marginal distributions. This time $S\leq T$ (requirement 2), but we have no way to control the correlation (requirement 1).
Is there a bivariate exponential distribution that meets both of my requirements?