copBasic (version 2.0.1)

COP: The Copula

Description

Compute the copula or joint distribution function through a copula as shown by Nelsen (2006, p. 18) is the joint probability

$$\mathrm{Pr}[U \le u, V \le v] = \mathbf{C}(u,v)\mbox{.}$$

The copula is an expression of the joint probability that both $U \le u$ and $V \le v$. A radial or permutation symmetric copula is one such that $\mathbf{C}(u,v) = \mathbf{C}(v,u)$ otherwise the copula is asymmetric. A copula is a type of dependence function that permits straightforward characterization of dependence from independence.

The copula inversions $t = \mathbf{C}(u{=}U, v)$ or $t = \mathbf{C}(u, v{=}V)$ for a given $t$ and $U$ or $V$ are provided by COPinv and COPinv2, respectively.

A copula exists in the domain of the unit square ($\mathcal{I}^2 = [0, 1]\times [0,1]$) and is a grounded function meaning that $$\mathbf{C}(u,0) = 0 = \mathbf{C}(0,v) \mbox{\ and\ thus\ } \mathbf{C}(0,0) = 0\mbox{, }$$ and other properties of a copula are that $$\mathbf{C}(u,1) = u \mbox{\ and\ } \mathbf{C}(1,v) = v\mbox{\ and}$$ $$\mathbf{C}(1,1) = 1\mbox{.}$$

Copulas can be combined with each other (convex2COP, composite1COP, composite2COP, and composite3COP) to form more complex and sophisticated dependence structures. Also copula multiplication---a special product of two copulas---yields another copula (see prod2COP).

Usage

COP(u, v, cop=NULL, para=NULL, reflect=c("cop", "surv", "acute", "grave"), ...)

Arguments

u

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

v

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

cop

A copula function with vectorization as in asCOP;

para

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

reflect

The reflection of the copula form (see Note) and the default "cop" is the usual copula definition (also see simCOPmicro); and

...

Additional arguments to pass to the copula.

Value

• Value(s) for the copula are returned.

References

Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.

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

See Also

coCOP, duCOP, surCOP, surfuncCOP

Examples

u <- runif(1); v <- runif(1)
COP(cop=W,u,v); COP(cop=P,u,v); COP(cop=M,u,v); COP(cop=PSP,u,v)

F <- 0.75 # 75th percentile, nonexceedance
G <- 0.20 # 25th percentile, nonexceedance
bF <- 1 - F; bG <- 1 - G  # exceedance

# What is the probability that both X and Y are less than
# 75th and 20th percentiles, respectively?
COP(cop=P,F,G)

# What is the probability that both X and Y are greater than
# 75th and 20th percentiles, respectively?
surCOP(cop=P,bF,bG)

# What is the probability that either X or Y are less than
# the 75th and 20th percentiles, respectively?
duCOP(cop=P,F,G)

# What is the probability that either X or Y are greater than
# the 75th and 20th percentiles, respectively?
coCOP(cop=P,bF,bG)

# Repeat for the PSP copula:
# What is the probability that both X and Y are less than
# 75th and 20th percentiles, respectively?
COP(cop=PSP,F,G)

# What is the probability that both X and Y are greater than
# 75th and 20th percentiles, respectively?
surCOP(cop=PSP,bF,bG)

# What is the probability that either X or Y are less than
# the 75th and 20th percentiles, respectively?
duCOP(cop=PSP,F,G)

# What is the probability that either X or Y are greater than
# the 75th and 20th percentiles, respectively?
coCOP(cop=PSP,bF,bG)

# Both of these summations equal unity
COP(cop=PSP,F,G)      + coCOP(cop=PSP,bF,bG)
surCOP(cop=PSP,bF,bG) + duCOP(cop=PSP,F,G)

F <- 0.99 # 99th percentile, nonexceedance
G <- 0.50 # 50th percentile, nonexceedance
bF <- 1 - F # nonexceedance
bG <- 1 - G # nonexceedance

# What is the probability that both X and Y are less than
# 99th and 50th percentiles, respectively?
COP(cop=P,F,G)

# What is the probability that both X and Y are greater than
# 99th and 50th percentiles, respectively?
surCOP(cop=P,bF,bG)

# What is the probability that either X or Y are less than
# the 99th and 50th percentiles, respectively?
duCOP(cop=P,F,G)

# What is the probability that either X or Y are greater than
# the 99th and 50th percentiles, respectively?
coCOP(cop=P,bF,bG)