I would like to simulate random variables where correlation is taken into account, but the random variables are not necessarily normally distributed. is there are way to measure correlation between two variables that are arbitrarily distributed? I have encountered the rank-correlation coefficient. Is that the best choice? Do you know of implementations that provide them?
you can use elliptical distributions http://en.wikipedia.org/wiki/Elliptical_distribution or copulas http://en.wikipedia.org/wiki/Copula_%28probability_theory%29
There is a R package 'fCopulae' which might help, for instance the 'rellipticalCopula' function generates correlated random samples from a variety of distributions.
hth.
Salih Çelebioğlu, Department of Statistics, Gazi University
As Eik Vettorazzi pointed out you can use elliptical distributions and use copulas, but I advise you to use copulas. Because copulas are more general than the elliptical distributions. Especially, Archimedean copulas are very easy to use among copulas, since they can be constructed very easily, as stated (Nelsen, 2006). Copulas are the most popular tools in these years. By separating into two parts, copulas makes easy to estimate in multivariate distributions. I see copulas as the skeleton of multivariate distributions. We can see the simulation in multivariate distributions as in two stages: first, the sample selection from marginal distributions and then simulation from underlying copulas. You can use the following references for the details on this subject.
-R. B. Nelsen, An Introduction to Copulas, Second Edition, Springer Verlag, 2006.
-D.D. Mari and S. Kotz, Correlation and Dependence, Imperial College Pres, 2004.
-H. Joe, Multivariate Models and Dependence Concepts, Chapman and Hall, 1997.
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