surCOP: The Survival Copula

Description

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

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

Value

Value(s) for the survival copula are returned.

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).

Author

W.H. Asquith

References