surfuncCOP: The Joint Survival Function

Description

Compute the joint survival function from a copula (Nelsen, 2006, p. 33), which is defined as $$\overline{\mathbf{C}}(u,v) = \mathrm{Pr}[U > u, V > v] = 1 - u - v + \mathbf{C}(u,v) = \hat\mathbf{C}(1-u, 1-v)\mbox{,}$$ where $\hat\mathbf{C}(u',v')$ is the survival copula (see surCOP), which is defined by $$\hat{\mathbf{C}}(u',v') = \mathrm{Pr}[U > u, V > v] = u' + v' - 1 + \mathbf{C}(1-u',1-v')\mbox{.}$$

Although the joint survival function is an expression of the probability that both $U > v$ and $U > v$, it is not a copula.

Usage

surfuncCOP(u, v, cop=NULL, para=NULL, ...)

Arguments

Nonexceedance probability $u$ in the $X$ direction;

Nonexceedance probability $v$ in the $Y$ direction;

cop

A copula function;

para

Vector of parameters or other data structure, if needed, to pass to the copula; and

...

Additional arguments to pass (such as parameters, if needed, for the copula in the form of a list.

Value

Value(s) for the joint survival function are returned.

References

Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.

Examples

Run this code

"MOcop.formula" <- function(u,v, para=para, ...) {
   alpha <- para[1]; beta <- para[2]; return(min(v*u^(1-alpha), u*v^(1-beta)))
}
"MOcop" <- function(u,v, ...) { asCOP(u,v, f=MOcop.formula, ...) }
u <- 0.2; v <- 0.75; ab <- c(1.5, 0.3)
# U **and** V are less than or equal to a threshold +
# U **or**  V are less than or equal to a threshold
surCOP(1-u,1-v, cop=MOcop, para=ab) + duCOP(u,v, cop=MOcop, para=ab) # UNITY
surfuncCOP(u,v, cop=MOcop, para=ab) + duCOP(u,v, cop=MOcop, para=ab) # UNITY
# The joint survival function is not a copula so it does not increases to the
# upper right with increasing exceedance probabilities. Let us show that by
# hacking the surCOP function into a copula for feeding back into the algorithmic
# framework of copBasic.
UsersCop <- function(u,v, para=NULL) {
     afunc <- function(u,v, theta=para) { surfuncCOP(u, v, cop=N4212cop, para=theta)}
     return(asCOP(u,v, f=afunc)) }
image(gridCOP(cop=UsersCop, para=1.15), col=terrain.colors(20),
      xlab="U, NONEXCEEDANCE PROBABILITY", ylab="V, NONEXCEEDANCE PROBABILITY")