simulating Gumbel’s bivariate exponential distribution

[This article was first published on R – Xi'an's Og, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

A challenge interesting enough for a sunny New Year morn, found on X validated, namely the simulation of a bivariate exponential distribution proposed by Gumbel in 1960, with density over the positive quadrant in IR²

{}^{ [(\lambda_2+rx_1)(\lambda_1+rx_2)-r]\exp[-(\lambda_1x_1+\lambda_2x_2+rx_1x_2)]}

Although there exists a direct approach based on the fact that the marginals are Exponential distributions and the conditionals signed mixtures of Gamma distributions, an accept-reject algorithm is also available for the pair, with a dominating density representing a genuine mixture of four Gammas, when omitting the X product in the exponential and the negative r in the first term. The efficiency of this accept-reject algorithm is high for r small. However, and in more direct connection with the original question, using this approach to integrate the function equal to the product of the pair, as considered in the original paper of Gumbel, is much less efficient than seeking a quasi-optimal importance function, since this importance function is yet another mixture of four Gammas that produces a much reduced variance at a cheaper cost!

To leave a comment for the author, please follow the link and comment on their blog: R – Xi'an's Og. offers daily e-mail updates about R news and tutorials about learning R and many other topics. Click here if you're looking to post or find an R/data-science job.
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

Never miss an update!
Subscribe to R-bloggers to receive
e-mails with the latest R posts.
(You will not see this message again.)

Click here to close (This popup will not appear again)