I am looking for formulas to calculate Kendall's tau ($\tau_\mathrm{C}$) from copula parameter (or vice versa). I am interested in extreme-t and Hüsler-Reiss copulas. It is not an issue if numerical integration is required, but I am expecting something simpler than the general formula:

$$ {\tau _{\rm{C}}} = 4 \cdot {\rm{E}}\left[ {C\left( {U,V} \right)} \right] - 1 = 4 \cdot \int\limits_0^1 {\int\limits_0^1 {C\left( {u,v} \right) \cdot {\rm{d}}C\left( {u,v} \right) - 1} } $$

$$ {\tau _{\rm{C}}} = 1 - 4 \cdot \int\limits_0^1 {\int\limits_0^1 {\frac{\partial }{{\partial u}}C\left( {u,v} \right) \cdot \frac{\partial }{{\partial v}}C\left( {u,v} \right)} } \cdot {\rm{d}}u \cdot {\rm{d}}v $$

As for Archimedean copulas the calculation can be simplified to a one-dimensional integral by using their generator function (see e.g. for Clayton copula), I hope for something similar for extreme copulas as well.

I've found the copula package that implemented extreme-t and Hüsler-Reiss copulas and functions to calculate Kendall's tau from parameter (tau()) and vice versa (iTau()). Unfortunately I was not able to extract from the source code how it calculates $\tau_\mathrm{C}$ from the copula parameter. I suspect that it interpolates between pre-computed values. I need the connection between $\tau_\mathrm{C}$ and copula parameter for $\tau_\mathrm{C}$ close to $1.0$ ($>0.98$). In this region the iTau() function of copula package gives infinite value for the parameter, which I suspect to be incorrect.



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