## simulating from the joint cdf

Posted in Books, Kids, pictures, R, Statistics, University life with tags , , , , , , , , on July 13, 2022 by xi'an

An X validated question (what else?!) brought back (to me) the question of handling a bivariate cdf for simulation purposes. In the specific case of a copula when thus marginals were (well-)known…. And led me to an erroneous chain of thought, fortunately rescued by Robin Ryder! When the marginal distributions are set, the simulation setup is indeed equivalent to a joint Uniform simulation from a copula

$\mathbb P[U_1\leq u_1,U_2\leq u_2,\dots,U_d\leq u_d]=C(u_1,u_2,\dots,u_d)$

In specific cases, as for instance the obvious example of Gaussian copulas, there exist customised simulation algorithms. Looking for more generic solutions, I turn to the Bible, where Chapter XI[an], has two entire sections XI.3.2. and XI.3.3 on the topic (even though Luc Devroye does not use the term copula there despite them being introduced in 1959 by A, Sklar, in response to a query of M. Fréchet). In addition to a study of copulas, both sections contain many specific solutions (as for instance in the [unnumbered] Table on page 585) but I found no generic simulation method. My [non-selected] answer to the question was thus to propose standard solutions such as finding one conditional since the marginals are Uniform. Which depends on the tractability of the derivatives of C(·,·).

However, being dissatisfied with this bland answer, I thought further about the problem and came up with a fallacious scheme, namely to first simulate the value p of C(U,V) by drawing a Uniform, and second simulate (U,V) conditional on C(U,V)=p. Going as far as running an R code on a simple copula, as shown above. Fallacious reasoning since (as I knew already!!!), C(U,V) is not uniformly distributed! But has instead a case-dependent distribution… As a (connected) aside, I wonder if the generator attached with Archimedean copulas has any magical feature that help with the generation of the associated copula.

## simulating correlated random variables [cont’ed]

Posted in Books, Kids, Statistics with tags , , , , on May 28, 2015 by xi'an

Following a recent post on the topic, and comments ‘Og’s readers kindly provided on that post, the picture is not as clear as I wished it was… Indeed, on the one hand, non-parametric measures of correlation based on ranks are, as pointed out by Clara Grazian and others, invariant under monotonic transforms and hence producing a Gaussian pair or a Uniform pair with the intended rank correlation is sufficient to return a correlated sample for any pair of marginal distributions by the (monotonic) inverse cdf transform.  On the other hand, if correlation is understood as Pearson linear correlation, (a) it is not always defined and (b) there does not seem to be a generic approach to simulate from an arbitrary triplet (F,G,ρ) [assuming the three entries are compatible]. When Kees pointed out Pascal van Kooten‘s solution by permutation, I thought this was a terrific resolution, but after thinking about it a wee bit more, I am afraid it is only an approximation, i.e., a way to return a bivariate sample with a given empirical correlation. Not the theoretical correlation. Obviously, when the sample is very large, this comes as a good approximation. But when facing a request to simulate a single pair (X,Y), this gets inefficient [and still approximate].

Now, if we aim at exact simulation from a bivariate distribution with the arbitrary triplet (F,G,ρ), why can’t we find a generic method?! I think one fundamental if obvious reason is that the question is just ill-posed. Indeed, there are many ways of defining a joint distribution with marginals F and G and with (linear) correlation ρ. One for each copula. The joint could thus be associated with a Gaussian copula, i.e., (X,Y)=(F⁻¹(Φ(A)),G⁻¹(Φ(B))) when (A,B) is a standardised bivariate normal with the proper correlation ρ’. Or it can be associated with the Archimedian copula

C(u; v) = (u + v − 1)-1/θ,

with θ>0 defined by a (linear) correlation of ρ. Or yet with any other copula… Were the joint distribution perfectly well-defined, it would then mean that ρ’ or θ (or whatever natural parameter is used for that copula) do perfectly parametrise this distribution instead of the correlation coefficient ρ. All that remains then is to simulate directly from the copula, maybe a theme for a future post…