Numerical Derivative Inverse of a Copula for U with respect to V
Numerical Derivative Inverse of a Copula for U with respect to V
Compute the inverse of a numerical partial derivative for U with respect to V of a copula, which is a conditional quantile function for nonexceedance probability t, or [REMOVE_ME]t=cv(u)=C1∣2(−1)(u∣v)=δvδC(u,v)\mbox,[REMOVEME2]
and solving for u. 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.
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Description
Compute the inverse of a numerical partial derivative for U with respect to V of a copula, which is a conditional quantile function for nonexceedance probability t, or
t=cv(u)=C1∣2(−1)(u∣v)=δvδC(u,v)\mbox,
and solving for u. 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.
v: A single nonexceedance probability v in the Y 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.
delv: The Δv interval for the derivative;
para: Vector of parameters or other data structure, if needed, to pass to cop; and
...: Additional arguments to pass to the copula.
Returns
Value(s) for the derivative inverse are returned.
References
Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.
Author(s)
W.H. Asquith
See Also
derCOP2
Examples
u <- runif(1); t <- runif(1)derCOPinv2(u,t, cop=W)# perfect negative dependencederCOPinv2(u,t, cop=P)# independencederCOPinv2(u,t, cop=M)# perfect positive dependencederCOPinv2(u,t, cop=PSP)# a parameterless copula example## Not run:# Simulate 500 values from product (independent) copulaplot(NA,NA, type="n", xlim=c(0,1), ylim=c(0,1), xlab="U", ylab="V")for(i in1:500){ v <- runif(1); t <- runif(1) points(derCOPinv2(cop=P, v, t),v, cex=0.5, pch=16)# black dots}# Simulate 500 of a Frechet Family copula and note crossing singularities.for(i in1:500){ v <- runif(1); t <- runif(1) u <- derCOPinv2(v, t, cop=FRECHETcop, para=list(alpha=0.7, beta=0.169)) points(u,v, cex=2, pch=16, col=2)# red dots}### End(Not run)