Normal copula vs Survival copula Can someone tell me the actual differences between the Survival copula and Normal copula model in terms of the programming aspects in R. Am working on bivariate dataset with censored observations and am having hard time differentiating in the code as well as the their behaviors with regards to different copula classes eg Archimedian like Gumbel, Frank and Clayton. How does the two types of copula differs in their results? and how are they related as well??. I know only that the differences is that Normal copula uses Marginal distributions while survival copula uses the survival functions but i dont know how to differentiate that more intuitively when it comes to Programming and interpretation. Any helps of document will be highly appreciated.
 A: I will discuss the differences and how they are related.
For two Gaussian random variables $X, Y$, the joint CDF, or joint probability, is
$$F(x,y) = P(X\leq x, Y \leq y)$$
Survival copulas instead ask the question what is
$$\bar{F}(x,y) = P(X> x, Y > y)?$$
which is linked to the first Gaussian copula formula by
$$\bar{F}(x,y) = P(X> x) + P(Y > y) - 1 + P(X\leq x, Y \leq y)$$
where $\bar{F}(x,y)$ is called survival probability in actuarial science.
In the same way that Sklar's theorem can recover a copula function $C(u,v)$ for the Gaussian case, Sklar still applies when recovering a survival copula function, $\hat{C}(u,v)$, such that
$$\bar{F}(x,y) = \hat{C}(1-F(x), 1-F(y))$$
The link between Gaussian and survival copula is
$$\hat{C}(u,v) = u + v - 1 + C(1-u,1-v)$$
Why is the survival copula useful? Because with regular copulas, the lower tail index of the joint tail dependence can be measured as $\lambda_L$ representing the association of major downside events, whereas the upper tail index $\lambda_U$ can be measured from survival copulas, representing major upside events. Copulas with Gaussian marginal probabilities and tail dependence, however, assume no asymmetry between downside and upside events.
Algorithms for computing survival probabilities and copulas can be found in programming texts.
