The earliest and the simplest known bivariate exponential distribution, introduced by Gumbel (1960), has joint survivor function and joint probability density function given by:

\begin{equation}\label{eq:sf} S(x, y) = \exp\left[- \left(\alpha\,x + \beta\,y + \theta\,\alpha\,\beta\,x\,y\right)\right] \end{equation} and \begin{equation}\label{eq:pdf} f(x, y) = \left[(1 - \theta)\,\alpha\,\beta + \theta\,\alpha^2\,\beta\,x + \theta\,\alpha\,\beta^2\,y + \theta^2\,\alpha^2\,\beta^2\,x\,y\right]\,S(x, y) \end{equation} respectively, where $ x, y >0 $, $ \alpha, \beta \geq 0 $ and $ 0 < \theta < 1 $.

I want to generate random samples from this distribution. It is important to point out that the marginal distributions of X and Y are exponential with parameters $\alpha$ and $\beta$.


1 Answer 1


Given the joint density\begin{equation} f(x, y) = \left[(1 - \theta)\,\alpha\,\beta + \theta\,\alpha^2\,\beta\,x + \theta\,\alpha\,\beta^2\,y + \theta^2\,\alpha^2\,\beta^2\,x\,y\right]\,S(x, y) \end{equation} the conditional density of $Y$ given $X=x$ is such that \begin{align} f(y|x) &\propto \left[(1 - \theta)\,\alpha\,\beta + \theta\,\alpha^2\,\beta\,x\right]\, S(x, y)+ \left[\theta\,\alpha\,\beta^2 + \theta^2\,\alpha^2\,\beta^2\,x\right]\,y\,S(x, y)\\ &\propto \left[1 - \theta+ \theta\,\alpha x\right]\, S(x, y)+ \left[\theta + \theta^2\,\alpha\,\beta\,x\right]\,y\,S(x, y)\\ &\propto \left[1 - \theta+ \theta\,\alpha x\right]\, \exp\left[- \left(\beta\,y + \theta\,\alpha\,\beta\,x\,y\right)\right]+ \left[\theta + \theta^2\,\alpha\,\beta\,x\right]\,y\,\exp\left[- \left( \beta\,y + \theta\,\alpha\,\beta\,x\,y\right)\right]\\ &\propto \left[1 - \theta+ \theta\,\alpha x\right]\, \exp\left[- \beta\,y\left( 1+ \theta\,\alpha\,x\right)\right]+ \left[\theta + \theta^2\,\alpha\,\beta\,x\right]\,y\,\exp\left[- \beta\,y\left( 1 + \theta\,\alpha x \right)\right]\\\end{align} which means that the conditional distribution of $Y$ given $X=x$ is a mixture of two distributions,

  1. the exponential $\mathcal{E}(\beta\,\left\{ 1+ \theta\,\alpha\,x\right\})$ distribution
  2. the Gamma $\mathcal{G}a(2,\beta\,\left\{ 1+ \theta\,\alpha\,x\right\})$ distribution

with respective weights $1$ and $$ \beta^{-1}\left[\theta + \theta^2\,\alpha\,\beta\,x\right]\times\left( 1 + \theta\,\alpha x \right)^{-2}$$up to normalisation. It is therefore straightforward to simulate $X$ marginally and then $Y$ given this realisation of $X$.


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