derCOPinv: Numerical Derivative Inverse of a Copula for V with respect to U

Description

Compute the inverse of a numerical partial derivative for $V$ with respect to $U$ of a copula, which is a conditional quantile function for nonexceedance probability $t$, or $$t = c_u(v) = \mathbf{C}^{(-1)}_{2|1}(v|u) = \frac{\delta \mathbf{C}(u,v)}{\delta u}\mbox{,}$$ and solving for $v$. Nelsen (2006, pp. 13, 40--41) shows that this inverse is quite important for random variable generation using the conditional distribution method. This function is not vectorized.

Usage

derCOPinv(cop=NULL, u, t, delu=.Machine$double.eps^0.50, para=NULL, ...)

Arguments

cop

A copula function;

A single nonexceedance probability $u$ in the $X$ direction;

A single nonexceedance probability level $t$;

delu

The $\Delta u$ interval for the derivative;

para

Vector of parameters or other data structures, if needed, to pass to cop; and

...

Additional arguments to pass to the copula.

Value

Value(s) for the derivative inverse are returned.

References

Durante, F., 2007, Families of copulas, Appendix C, in Salvadori, G., De Michele, C., Kottegoda, N.T., and Rosso, R., 2007, Extremes in Nature---An approach using copulas: Springer, 289 p.

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

Johnson, M.E., 1987, Multivariate statistical simulation: New York, John Wiley, 230 p.

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

Examples

Run this code

u <- runif(1); t <- runif(1)
derCOPinv(u,t, cop=W)   # perfect negative dependence
derCOPinv(u,t, cop=P)   # independence
derCOPinv(u,t, cop=M)   # perfect positive dependence
derCOPinv(u,t, cop=PSP) # a parameterless copula example
# Simulate 500 values from product (independent) copula
plot(NA,NA, type="n", xlim=c(0,1), ylim=c(0,1), xlab="U", ylab="V")
for(i in 1:500) {
   u <- runif(1); t <- runif(1)
   points(u,derCOPinv(cop=P,u,t), cex=0.5, pch=16) # black dots
}
# Now simulate 500 from the Nelsen 4.2.12 copula.
for(i in 1:500) {
   u <- runif(1); t <- runif(1)
   points(u,derCOPinv(cop=N4212cop,para=9.3,u,t), cex=2, pch=16, col=2) # red dots
}