kfuncCOPinv {copBasic}

The Inverse Kendall Function of a Copula

Description

Compute the (numerical) inverse F^{(-1)}_K(z) \equiv z(F_K) of the Kendall Function F_K(z; \mathbf{C}) (kfuncCOP) of a copula \mathbf{C}(u,v) given nonexceedance probability F_K. The z is the joint probability of the random variables U and V coupled to each other through the copula \mathbf{C}(u,v) and the nonexceedance probability of the probability z is F_K—statements such as “probabilities of probabilities” are rhetorically complex so pursuit of word precision is made herein.

Usage

kfuncCOPinv(f, cop=NULL, para=NULL, subdivisions=100L,
               rel.tol=.Machine$double.eps^0.25, abs.tol=rel.tol, ...)

Arguments

f

Nonexceedance probability (0 \le F_K \le 1);

cop

A copula function;

para

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

subdivisions

Argument of same name passed to integrate() through kfuncCOP,

rel.tol

Argument of same name passed to integrate() through kfuncCOP,

abs.tol

Argument of same name passed to integrate() through kfuncCOP, and

...

Additional arguments to pass.

Value

The value(s) for z(F_K) are returned.

Note

The L-moments of Kendall Functions appear to be unresearched. Therefore, the kfuncCOPlmom and kfuncCOPlmoms functions were written. These compute L-moments on the CDF F_K(z) and not the quantile function z(F_K) and thus are much faster than trying to use kfuncCOPinv in the more common definitions of L-moments. A demonstration of the mean (first L-moment) of the Kendall Function numerical computation follows:

  # First approach
  "afunc" <- function(f) kfuncCOPinv(f, cop=GHcop, para=pi)
  integrate(afunc, 0, 1) # 0.4204238 with absolute error < 2.5e-05
  # Second approach
  kfuncCOPlmom(1, cop=GHcop, para=pi)  # 0.4204222

where the first approach uses z(F_K), whereas the second method uses integration for the mean on F_K(z).

Author(s)

W.H. Asquith

References

Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978–146350841–8.

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

See Also

kfuncCOP

Examples

## Not run: 
Z <- c(0,0.25,0.50,0.75,1) # Joint probabilities of a N4212cop
kfuncCOPinv(kfuncCOP(Z, cop=N4212cop, para=4.3), cop=N4212cop, para=4.3)
# [1] 0.0000000 0.2499984 0.5000224 0.7500112 1.0000000
## End(Not run)