Table 1.

Canonical choices of K, g and h_k(t) for Archimedean copulas that can be represented in the regression form of (5), i.e. $g(\mathrm{\Psi }(t))={\sum }_{k=1}^K{\alpha }_k(\theta )h_k(t)$. The last four copulas are labeled according to their indices in Table 4.1 in Nelsen (2006).

Name	K	g(Ψ(t))	hk(t)	αk
Ali-Mikhail-Haq	2	$\frac{1-\mathrm{\Psi }(t)}{\mathrm{\Psi }(t)}$	$h_1(t)=\frac{1-t}{t}$, h2(t) = h1(t)2	α1 = 2, α2 = (1 − θ)
Gumbel-Hougaard	1	log Ψ(t)	h1(t) = logt	α1 = 21/θ
Gumbel-Barnett	2	log Ψ(t)	h1(t) = logt, h2(t) = (h1(t))2	α1 = 2, α2 = −θ
(4.2.2) in Nelsen	1	1 − Ψ(t)	h1(t) = 1 − t	α1 = 21/θ
(4.2.7) in Nelsen	2	1 − Ψ(t)	h1(t) = 1 − t, h2(t) = h1(t)2	α1 = 2, α2 = −θ
(4.2.12) in Nelsen	2	$log\frac{\mathrm{\Psi }(t)}{1-\mathrm{\Psi }(t)}$	h1(t) = 1, $h_2(t)=log\frac{t}{1-t}$	${\alpha }_1=-\frac{log2}{\theta }$, α2 = 1
(4.2.18) in Nelsen	2	$\frac{1}{1-\mathrm{\Psi }(t)}$	h1(t) = 1, $h_2(t)=\frac{1}{1-t}$	${\alpha }_1=-\frac{log2}{\theta }$, α2 = 1