copBasic (version 1.7.1)

surCOP: The Survival Copula

Description

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

$$\hat{\mathbf{C}}(1-u,1-v)=\hat{\mathbf{C}}(u^{\prime },v^{\prime })=\mathrm{Pr}[U>u,V>v]=u^{\prime }+v^{\prime }-1+\mathbf{C}(1-u^{\prime },1-v^{\prime })\text{,}$$ where $u'$ and $v'$ are exceedance probabilities and $\mathbf{C}(u,v)$ is the copula.

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 $\overline{\mathbf{C}}(u,v)$ defined as (Nelsen, 2006, p. 33) $$\bar{\mathbf{C}}(u,v)=\mathrm{Pr}[U>u,V>v]=1-u-v+\mathbf{C}(u,v)\text{.}$$

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 a list.

Value

• Value(s) for the survival copula are returned.

References

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

See Also

COP, coCOP, duCOP, surfuncCOP

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
surfuncCOP(u, v, cop=PSP)                   # 0.4043928
# All three examples show joint prob. that U > u and V > v.

# 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")