copBasic (version 1.5.1)

qua.regressCOP2: Perform Quantile Regression using a Copula by Numerical Derivative Method for U with respect to V

Description

Perform quantile regression using a copula by numerical derivatives of the copula. If $X$ and $Y$ are random variables having quantile functions $x(F)$ and $y(G)$ and letting $x=\tilde{x}(y)$ denote a solution to $\mathrm{Pr}[X \le x\mid Y = y] = F$, where $F$ is a nonexceedance probability. The the curve $x=\tilde{x}(y)$ is the quantile regression curve of $U$ or $X$ with respect to $V$ or $Y$, respectively. If $F=1/2$, then median regression is performed (med.regressCOP2). Using copulas, the quantile regression is expressed as $$\mathrm{Pr}[X\le x\mid Y=y]=\mathrm{Pr}[U\le F(x)\mid V=F]=\mathrm{Pr}[U\le u\mid V=F]=\frac{\delta \mathbf{C}(u,v)}{\delta v}\text{,}$$ where $v = G(y)$ and $u = F(x)$. The general algorithm is

1. Set$\delta \mathbf{C}(u,v)/\delta v = F$,
2. Solve the regression curve$u = \tilde{u}(v)$(accomplished byderCOPinv2), and
3. Replace$u$by$x(u)$and$v$by$y(v)$.

The last step is optional as step two produces the regression in probability space, which might be desired, and step 3 actually transforms the probability regressions into the quantiles of the respective random variables.

Usage

qua.regressCOP2(F=0.5,
                V=seq(0.01,0.99, by=0.01),
                cop=NULL, para=NULL, ...)

Arguments

F

The nonexceedance probability $F$ to perform regression at,

V

A vector of $v$ nonexceedance probabilities,

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

• A data frame of the simulated values is returned.

References

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

See Also

med.regressCOP2, derCOPinv2

Examples

# Specify a negatively associated Plackett copula and perform the
# regression using the defaults. Plot the regression---median
# in this case
theta <- 0.10
R <- qua.regressCOP2(cop=PLACKETTcop, para=c(theta))
plot(R$U,R$V, type="l", lwd=3, xlim=c(0,1), ylim=c(0,1))
lines(R$U,(1+(theta-1)*R$U)/(theta+1), col=2)

R <- qua.regressCOP2(F=0.90,cop=PLACKETTcop, para=c(theta))
lines(R$U,R$V, col=2, lwd=2)

R <- qua.regressCOP2(F=0.10,cop=PLACKETTcop, para=c(theta))
lines(R$U,R$V, col=2, lty=2)

# Specify a composite copula as two Placketts with respective
# parameters and then the mixing parameters alpha and beta.
para <- list(cop1=PLACKETTcop,  cop2=PLACKETTcop,
             para1=c(0.14),    para2=c(21),
             alpha=0.04,        beta=0.68)
# Initial a plot
plot(c(0,1),c(0,1), type="n", lwd=3,
     xlab="U, NONEXCEEDANCE PROBABILITY",
     ylab="V, NONEXCEEDANCE PROBABILITY")
# Draw the regression of V on U and then U on V (swap=TRUE)
qua.regressCOP.draw(cop=composite2COP, para=para)
qua.regressCOP.draw(cop=composite2COP, para=para, swap=TRUE, lty=2)