# composition versus inversion

**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.

**W**hile trying to convey to an OP on X validated why the inversion method was not always the panacea in pseudo-random generation, I took the example of a mixture of K exponential distributions when K is very large, in order to impress (?) upon said OP that solving F(x)=u for such a closed-form cdf F was very costly even when using a state-of-the-art (?) inversion algorithm, like uniroot, since each step involves adding the K terms in the cdf. Selecting the component from the cumulative distribution function on the component proves to be quite fast since using the rather crude

x=rexp(1,lambda[1+sum(runif(1)>wes)])

brings a 100-fold improvement over

Q = function(u) uniroot((function(x) F(x) - u), lower = 0, upper = qexp(.999,rate=min(la)))[1] #numerical tail quantile x=Q(runif(1))

when K=10⁵, as shown by a benchmark call

test elapsed 1 compo 0.057 2 Newton 45.736 3 uniroot 5.814

where Newton denotes a simple-minded Newton inversion. I wonder if there is a faster way to select the component in the mixture. Using a while loop starting from the most likely components proves to be much slower. And accept-reject solutions are invariably slow or fail to work with such a large number of components. Devroye’s Bible has a section (XIV.7.5) on simulating sums of variates from an infinite mixture distribution, but, for once, nothing really helpful. And another section (IV.5) on series methods, where again I could not find a direct connection.

**leave a comment**for the author, please follow the link and comment on their blog:

**R – Xi'an's Og**.

R-bloggers.com 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.