I'm currently doing some reading on copula function. For a bivariate cdf, by Sklar's theorem, there exists a copula function,C, such that
$P(U<u,V<v) = C(u, v)$.
As written by Nelsen (2006), A survival copula is given by
$P(U>u,V>v) = 1-u-v+C(u,v)$.
From here, implying 1-u=u* and 1-v=v* the left hand side can be simplified as
$P(U>u,V>v)=Ĉ(u^*,v^*)=u^*+v^*-1+C(1-u^*,1-v^*)$.
Here, the link between ordinary copula (C without hat) and survival copula (C with hat) is understandable. However, I have read a recent journal regarding semi-survival copula which stated that for a semi-survival copula, it is written as
$P(U<u,V>v) = Ĉ(u,v^*) = C(u,1-v)$.
I tried to derive the semi-survival using the same concept as survival copula but get a different equation from above. The derivation goes as follows:
$Ĉ(u,1-v) = 1-(1-u+C(u,v))$
$Ĉ(u,1-v) = u-C(u,v)$
Let 1-v=v*
This implies $Ĉ(u,v^*) = u-C(u,1-v^*)$.
I got an extra u and the C(u,1-v*) has negative sign. Which part did I do wrong?
Relations like that can be derived in a straightforward manner without looking into any literature from $$\tag{1} \mathbb P(U<u,V<v)=C(u,v)\,, $$ and \begin{align} 1&=\underbrace{\mathbb P(U<u,V<v)}_{C(u,v)}+\mathbb P(U\ge u,V<v)+\mathbb P(U<u,V\ge v)\nonumber\\ &\quad+\mathbb P(U\ge u,V\ge v)\,.\tag{2} \end{align} The last term is what seems to be called survival copula. The middle terms are \begin{align}\tag{3} \mathbb P(U\ge u,V<v)&=\mathbb P(V<v)-\mathbb P(U<u,V<v)=C(+\infty,v)-C(u,v)\,,\\ \tag{4} \mathbb P(U<u,V\ge v)&=\mathbb P(U<u)-\mathbb P(U<u,V<v)=C(v,+\infty)-C(u,v)\,. \end{align}
For $$\tag{5} \mathbb P(U<u,V\ge v)=C(u,1-v) $$ to hold we must have $$\tag{6} \mathbb P(U<u,V\ge v)=\mathbb P(U<u,V<1-v)\,. $$ In particular, $$\tag{7} \mathbb P(V\ge v)=1-\mathbb P(V<v)=\mathbb P(V<1-v)\,. $$ Taking the derivative w.r.t. $v$ shows that the density of $V$ must satisfy $$\tag{8} p(v)=p(1-v)\,. $$ This means that $p$ must be symmetric around $\frac{1}{2}\,.$ Obviously, not every PDF satisfies this relationship. Two prime examples of such a density are that of the uniform distribution on $[0,1]$ or the standard normal PDF with mean $\frac{1}{2}\,:$ $$ p(v)=\frac{1}{\sqrt{2\pi}}\exp\Big(-\frac{(v-1/2)^2}{2}\Big)\,. $$