copBasic (version 2.0.1)

densityCOP: Density of a Copula

Description

The function numerically estimates the copula density for a sequence of $u$ and $v$ probabilities for which each sequence has equal steps that are equal to $\Delta(uv)$. The density $c(u,v)$ of a copula $\mathbf{C}(u,v)$ can be computed by $$c(u,v) = [\mathbf{C}(u_2,v_2) - \mathbf{C}(u_2,v_1) - \mathbf{C}(u_1,v_2) + \mathbf{C}(u_1,v_1)]/[\Delta(uv)\times\Delta(uv)]\mbox{,}$$ where $c(u,v) \ge 0$ (see Nelsen, 2006, p. 10; densityCOPplot). The joint density can be defined by the coupla density for continuous variables and is the ratio of the joint density funcion $f(x,y)$ for random variables $X$ and $Y$ to the product of the marginal densities ($f_x(x)$ and $f_y(y)$): $$c(F_x(x), F_y(y)) = \frac{f(x,y)}{f_x(x)f_y(y)}\mbox{,}$$ where $F_x(x)$ and $F_y(y)$ are the respective cumulative distribution functions of $X$ and $Y$, and lastly $u = F_x(x)$ and $v = F_y(y)$.

Usage

densityCOP(u,v, cop=NULL, para=NULL, deluv=.Machine$double.eps^0.25,
                truncate.at.zero=TRUE, ...)

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;

deluv

The change in the sequences ${u, v} \mapsto \delta, \ldots, 1-\delta; \delta = \Delta(uv)$ probabilities;

truncate.at.zero

A density must be $c(u,v) \ge 0$, but because this computation is based on a rectangular approximation and not analytical, there exists a possibility that very small rectangles could result in numerical values in Rthat are less than zero and blamed on rou

...

Additional arguments to pass to the copula function.

Value

• Value(s) for $c(u,v)$ are returned.

encoding

utf8

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

simCOP, densityCOPplot, kullCOP

Examples

# Joe (2014, p. 164) shows the closed form copula density for the Plackett.
"dPLACKETTcop" <- function(u,v,para) {
   uv <- u*v; upv <- u + v; eta <- para - 1
   A <- para*(1+eta*(upv - 2*uv)); B <- ((1+eta*upv)^2 - 4*para*eta*uv)^(3/2)
   return(A/B)
}
dPLACKETTcop(0.32, 0.74, para=1.3)                # 0.9557124
densityCOP(0.32, 0.74, cop=PLACKETTcop, para=1.3) # 0.9557153

# Joe (2014, p. 170) shows the closed form copula density for "Bivariate Joe/B5" copula
"JOEB5cop" <- function(u,v,para) {
   up <- (1-u)^para; vp <- (1-v)^para
   return(1 - (up + vp - up*vp)^(1/para))
}
"dJOEB5cop" <- function(u,v,para) {
   up <- (1-u)^para; vp <- (1-v)^para; eta <- para - 1
   A <- (up + vp - up*vp); B <- (1-u)^eta * (1-v)^eta; C <- (eta + A)
   return(A^(-2 + 1/para) * B * C)
}
densityCOP(0.32, 0.74, cop=JOEB5cop, para=1.3) # 0.9410653
dJOEB5cop(0.32, 0.74, para=1.3)                # 0.9410973