Implemented copulas and the corresponding derivative limit,
as \(\theta \to a\), where \(a\) is such that \(C_a(u,v)=uv\).
An estimate for \(\theta\) is obtained based on the copula function used.
and the derivatives are used to obtain an estimate for \(K_1\).
The functions ‘frank’, ‘amh’, ‘fgm’ and ‘gauss’
are shortcuts for copula::frankCopula(),
copula::amhCopula(), copula::fgmCopula() and
copula::normalCopula() from package ‘copula’,
respectively.
frankdCtheta_frank(u, v)
amh
dCtheta_amh(u, v)
fgm
dCtheta_fgm(u, v)
gauss
dCtheta_gauss(u, v)
Archimedean copula objects of class ‘frankCopula’, ‘amhCopula’ or a
Farlie-Gumbel-Morgenstern copula object of class ‘fgmCopula’ or an elliptical
copula object of class ‘normalCopula’. For details, see
archmCopula, fgmCopula and
ellipCopula.
The derivative functions return the limit, as \(\theta \to 0\), of the derivative with respect to \(\theta\), corresponding to the copula functions.
An object of class frankCopula of length 1.
An object of class amhCopula of length 1.
An object of class fgmCopula of length 1.
An object of class normalCopula of length 1.
a real number between 0 and 1.
a real number between 0 and 1.
The constant \(K_1\) is given by
$$K_1 = \int_0^1\int_0^1\frac{1}{l_0(u)l_n(v)}\lim_{\theta\rightarrow a}\frac{\partial C_{\theta}(u,v)}{\partial\theta}\,dudv, $$
where \(I=[0,1]\), \(l_m(x):= F_m'\big(F_m^{(-1)}(x)\big)\) and \(\{F_n\}_{n \geq 0}\) is a sequence of absolutely continuous distribution functions