Design automatic soundness test for all copulas.

[lrnv]

The idea is that a few basic checks should exists and be the same for every copulas. In these tests, we might include :

• Marginal uniformity, to ensure that the model is indeed a copula. This can include verifying that the cdf has uniform margins AND that marginals samples are uniformely distributed (so a check on the cdf and a check on the samples). We alreayd have this one but we could generlize it a bit more and make it automatic

• check mandatory values of pdf&cdf on the frontier of the hypercube

• check that cdf is in [0,1] and that pdf is positive on some random points

• check numerically that the pdf is indeed the derivative of the cdf indeed

• check that tau \circ tau_inv is the identity on the domain

• check that rho \circ rho_inv is identitdy in the domain

• check that fitting the model works under a sklardist construct

• Archimedean have specific tests themselves.

• Generators could be checked numerically to be indeed monotone enough (compute theoretically the derivatives up to d-2 for exemple and check signs).

Some of these checks need to filter the models on the ones that implements a certain method... this looks complicated to do and i do not really want to do it by hand, but it is the right thing to do i think. Other ideas on stuff that could be automaticcally checked ?