surCOP: The Survival Copula

In wasquith/copBasic: General Bivariate Copula Theory and Many Utility Functions

The Survival Copula

Description

Compute the survival copula from a copula (Nelsen, 2006, pp. 32–34), which is defined as

\hat{\mathbf{C}}(1-u,1-v) = \hat{\mathbf{C}}(u',v') = \mathrm{Pr}[U > u, V > v] = u' + v' - 1 + \mathbf{C}(1-u', 1-v')\mbox{,}

where u' and v' are exceedance probabilities and \mathbf{C}(u,v) is the copula (COP). The survivial copula is a reflection of both U and V.

The survival copula is an expression of the joint probability that both U > v and U > v when the arguments a and b to \hat{\mathbf{C}}(a,b) are exceedance probabilities as shown. This is unlike a copula that has U \le u and V \le v for nonexceedance probabilities u and v. Alternatively, the joint probability that both U > u and V > v can be solved using just the copula 1 - u - v + \mathbf{C}(u,v), as shown below where the arguments to \mathbf{C}(u,v) are nonexceedance probabilities. The later formula is the joint survival function \overline{\mathbf{C}}(u,v) (surfuncCOP) defined for a copula (Nelsen, 2006, p. 33) as

\overline{\mathbf{C}}(u,v) = \mathrm{Pr}[U > u, V > v] = 1 - u - v + \mathbf{C}(u,v)\mbox{.}

Users are directed to the collective documentation in COP and simCOPmicro for more details on copula reflection.

Usage

surCOP(u, v, cop=NULL, para=NULL, exceedance=TRUE, ...)

Arguments

u	Exceedance probability u' = 1 - u (u nonexceedance based on exceedance) in the X direction;
v	Exceedance probability v' = 1 - v (v nonexceedance based on exceedance) in the Y direction;
cop	A copula function;
para	Vector of parameters or other data structure, if needed, to pass to the copula;
exceedance	A logical affirming whether u and v are really in exceedance probability or not? If FALSE, then the complements of the two are made internally and the nonexceedances can thus be passed; and
...	Additional arguments to pass (such as parameters, if needed, for the copula in the form of an R list).

Value

Value(s) for the survival copula are returned.

Note

The author (Asquith) finds the use of exceedance probabilities delicate in regards to Nelsen's notation. This function and coCOP have the exceedance argument to serve as a reminder that the survival copula as usually defined uses exceedance probabilities as its arguments.

Author(s)

W.H. Asquith

References

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

See Also

COP, coCOP, duCOP, surfuncCOP, simCOPmicro

Examples

u  <-  0.26; v  <- 0.55   # nonexceedance probabilities
up <- 1 - u; vp <- 1 - v  #    exceedance probabilities
surCOP(up, vp,   cop=PSP, exceedance=TRUE)  # 0.4043928
surCOP(u, v,     cop=PSP, exceedance=FALSE) # 0.4043928 (same because of symmetry)
surfuncCOP(u, v, cop=PSP)                   # 0.4043928
# All three examples show joint prob. that U > u and V > v.

## Not run: 
# A survival copula is a copula so it 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) { surCOP(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, EXCEEDANCE PROBABILITY", ylab="V, EXCEEDANCE PROBABILITY") #
## End(Not run)

wasquith/copBasic documentation built on Dec. 13, 2024, 6:39 p.m.