pdf/logpdf for Archimedean copula about copulas.jl

[lrnv]

What I remember without taking a look at it is that the number of derivatives you need is more or less the dimension of the model. Therefore, my ambition was to use ForwardDiff for the default implementation on ArchimedeanCopulas, as you point out. Feel free to implement it and PR if you want, otherwise I'll do it this summer. No need to stop arbitrarily at d=6, I plan to just let it be generic.

[mlkrock]

I just made a PR but I stopped at d=9, so can already say it's better than R there, but I'm not sure of a way to make it a derivative for general dimension d.

Edit: This PR result can be a problem in some cases, for example
pdf(JoeCopula(2,5.5),[.999;.999])
returns NaN, real value is approximately 1276.1015874698915.
I believe the problem is from algebraic cancellations that occur with ϕ and ϕ⁻¹ and these are not captured in the generic AD version of this pdf/logpdf.

JoeCopula generators seem to be a problem. Also getting several 1's when generating from JoeCopula (but not when the parameter is smaller)
findall(isone.(rand(JoeCopula(2,5.5),4000)))

[lrnv]

Hey ! I commented on the PR, let's discuss the implementation details there.

The Joe problem should maybe become a test case ? I agree with you that this is not satisfactory, as 1/1000 quantiles are sometimes needed when dealing with extremal problems (and therefore should be accurate).