copBasic (version 2.1.5)

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 and will not be so.

Usage

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

Arguments

cop

A copula function;

u

A single nonexceedance probability \(u\) in the \(X\) direction;

t

A single nonexceedance probability level \(t\);

trace

A logical controlling a message on whether the signs on the uniroot are the same---this is helpful in exploring the numerical derivative limits of a given implementation of a copula.

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.

See Also

derCOP

Examples

# NOT RUN {
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
# }
# NOT RUN {
# 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
} #
# }