densityCOP: Density of a Copula

Description

Numerically estimate 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 $\mathbf{C}(u,v)$ is numerically estimated by $$c(u,v) = \bigl[\mathbf{C}(u_2,v_2) - \mathbf{C}(u_2,v_1) - \mathbf{C}(u_1,v_2) + \mathbf{C}(u_1,v_1)\bigr]\, /\, \bigl[\Delta(uv)\times\Delta(uv)\bigr]\mbox{,}$$ 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 $f_y(y)$):

$$c\bigl(F_x(x), F_y(y)\bigr) = \frac{f(x,y)}{f_x(x)f_y(y)}\mbox{,}$$

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, the.zero=0, sumlogs=FALSE, ...)

Value

Value(s) for $c(u,v)$ are returned.

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 R that are less than zero. This usually can be blamed on rounding. This logical if TRUE truncates computed densities to zero, and the default assumes that the user is providing a proper copula. A FALSE value is used by the function isfuncCOP;
the.zero: The value for “the zero” where a small number might be useful for pseudo-maximum likelihood estimation using sumlogs;
sumlogs: Return the $\sum{\log c(u,v; \Theta)}$ where $\Theta$ are the parameters in para and this feature is provided for mleCOP; and
...: Additional arguments to pass to the copula function.

Author

W.H. Asquith

References

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