Compute the lower- and upper-tail orders (if they exist), respectively, of a copula $\mathbf{C}(u,v)$ according to Joe (2015, pp. 67--70). The tail order is a concept for the strength of dependence in the joint tails of the multivariate distribution. The opposing tails can be compared to assess tail order reflection symmetry (term by Joe (2015) for Nelsen's (2006) radial symmetry). Joe (2015) provides extensively analytical details but sufficient for the copBasic package, the tail orders can be numerically explored.The lower-tail order maybe numerically approximated by
$$\kappa^L_\mathbf{C} = \frac{\log(\mathbf{C}(t,t))}{\log(t)}\mbox{,}$$
for some small positive values of $t$, and similarly the upper-tail order maybe numerically approximated by
$$\kappa^U_\mathbf{C} = \frac{\log(\hat\mathbf{C}(t,t))}{\log(t)}\mbox{,}$$
where $\hat\mathbf{C}(u,v)$ is the survial copula (surCOP). Joe (2015) has potentially(?) conflicting notation in the context of the upper-tail order. The term reflection is used (p. 67) and lower tail order of the reflected copula is the same as the upper tail order of the original copula (p. 69). But Joe (2015, p. 67 only) uses the joint survival function (surfuncCOP) in the definition of $\kappa^U_\mathbf{C}$. As a note, the author of this package was not able to get tailordCOP to function properly for the upper-tail order using the joint survival function as implied on the bottom of Joe (2015, p. 67) and fortunately the fact that reflection is used in other contexts and used in analytical examples, the tailordCOP function uses the lower-tail order of the reflection (survival copula). Joe (2015) author also defines tail order parameter $\Psi$ but that seems to be a result of analytics and not implemented in this package. Lastly, the tail orders are extendable into $d$ dimensions, but only a bivariate ($d = 2$) is provided in this package.
The tail orders have various classifications for $\kappa = \kappa_L = \kappa_U$:
[object Object],[object Object],[object Object]
Joe (2015) provides additional properties:
[object Object],[object Object],[object Object]