kfuncCOPinv: The Inverse Kendall Function of a Copula

In copBasic: General Bivariate Copula Theory and Many Utility Functions

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, ...)

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; 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 N412cop
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)

copBasic documentation built on Oct. 17, 2023, 5:08 p.m.