Copula derivatives wrt arguments and parameters

[tnagler]

At some point we should have this.

Useful for

• faster MLE.
• computation of Fisher Information and similar quantities.
• Goodness-of-Fit tests

Requirement

In my opinion, we should ensure that it works for all parametric families.

Implementation options

• Numerical derivatives: super easy to implement, but slow. In particular, there will be no speed-up for (sequential) MLE. Could be a preliminary solution.
• Symbolic derivatives:
  • The "VineCopula approach" of copy&pasting generated code: Allows to enhance stability by manual modifications, but adds a lot of ugly code to the library.
  • At compile time in C++ using this or that: Very elegant and probably very little overhead at compile time, but doesn't allow manual modifications. We would need to ensure stability by a "hack" like truncation.
• Automatic differentiation: Also rather elegant, but heavy memory load at compile time. Possible frameworks: Eigen and Stan.

[tvatter]

• Numerical derivatives: we should have this as a benchmark, but I wouldn't release this as a feature.
• Symbolic derivatives:
  • The "VineCopula approach" of copy&pasting generated code: no way.
  • At compile time in C++ using this or that: I really like the codet solution. C++14 is not a good idea I think.
• Automatic differentiation: slow (and really stan is too heavy for this I think).