surfuncCOP {copBasic}

The Joint Survival Function

Description

Compute the joint survival function for 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 (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, \overline{\mathbf{C}}(u,v) is not a copula.

Usage

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

Arguments

u

Nonexceedance probability u in the X direction;

v

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.

Author(s)

W.H. Asquith

References

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

See Also

surCOP

Examples

"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

## Not run: 
# 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") #
## End(Not run)

## Not run: 
# Conditional return period (Salvadori et al., 2007, p. 159)
UV <- simCOP(n=100000, cop=PLACKETTcop, para=5, graphics=FALSE)
u <- 0.5; v <- 0.99; cd <- UV$V[UV$U > u]
by.counting <- length(cd[cd > v]) / length(cd)                  # 0.0172
by.theo     <- surfuncCOP(u,v, cop=PLACKETTcop, para=5) / (1-u) # 0.0166
by.ec       <- surfuncCOP(u,v, cop=EMPIRcop, para=UV)   / (1-u) # 0.0189
print(1/by.theo) # conditional return period for V > 0.99 given U > 0.5 
## End(Not run)