# Bivariate exponential distribution $(S, T)$ with controllable correlation and $S\leq T$

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:

1. $$\rho(S, T)$$ can be controlled by some parameter of the joint distribution
2. $$S\leq T$$

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?

Consider $$E \sim \mathcal{E}( \lambda_S - \lambda_C)$$ and $$C \sim \mathcal{E}(\lambda_C)$$. Now define \begin{align*} S & = \min \{ E , C \} \\ T & = \min \{ a E , C\} \end{align*} where $$a = \frac{ \lambda_S - \lambda_C}{\lambda_T - \lambda_C} > 1 .$$

You get $$S \sim \mathcal {E}(\lambda_S)$$ and $$T \sim \mathcal{E} (\lambda_T)$$, by construction $$S \leq T$$ and you have some ability to control correlation between two variables. Analytically, $$\rho(S,T) = \frac{\lambda_S+\lambda_C(1-\lambda_T/\lambda_S-\lambda_S/\lambda_T)}{\lambda_S-\lambda_C} .$$ If $$\lambda_C \approx 0$$ you get $$S \approx E$$ and $$S\approx a E$$ hence $$\rho \approx 1$$, but if $$\lambda_C \approx \lambda_T$$ then $$T \approx C$$ and $$S \approx \min\{E, C\}$$, meaning $$\rho\approx\lambda_T/\lambda_S$$.

With the following simulation I obtained an estimation of $$0.625$$ for the correlation with $$\lambda_S = 2$$, $$\lambda_T = 1.1$$ and $$\lambda_C = 1$$ (analytically we know $$\rho(S,T)=\frac{139}{220}\approx 0.632$$).

set.seed(1234)
lambda_S = 2; lambda_T = 1.1; lambda_C = 1
a = (lambda_S -lambda_C)/(lambda_T -lambda_C)

E = rexp(10000,  lambda_S - lambda_C)
C = rexp(10000,  lambda_C)

S =  apply(rbind(E,C), 2, min)
TT = apply(rbind( a*E,C), 2, min)

cor(S, TT)
# 0.6254737


Changing $$\lambda_S = 10$$ gives $$\rho(S,T)=\frac{1979}{9900}\approx 0.200$$.