qua.regressCOP2 {copBasic}

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

Description

Perform quantile regression (Nelsen, 2006, pp. 217–218) using a copula by numerical derivatives of the copula (derCOPinv2). 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. Then 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}\mbox{,}

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) (provided by derCOPinv2), 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

A single value of nonexceedance probability F to perform regression at and defaults to median regression F=1/2;

v

Nonexceedance probability v in the Y direction;

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

An R data.frame of the regressed probabilities of U and V=v is returned.

Author(s)

W.H. Asquith

References

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

See Also

med.regressCOP2, derCOPinv2

Examples

## Not run: 
# Use a positively associated Plackett copula and perform quantile regression
theta <- 0.10
R <- qua.regressCOP2(cop=PLACKETTcop, para=theta) # 50th percentile regression
plot(R$U,R$V, type="l", lwd=6, xlim=c(0,1), ylim=c(0,1), col=8)
lines((1+(theta-1)*R$V)/(theta+1),R$V, col=4, lwd=1) # theoretical for Plackett,
# compare the theoretical form to that in qua.regressCOP---just switch terms around
# because of symmetry
R <- qua.regressCOP2(f=0.90, cop=PLACKETTcop, para=theta) # 90th-percentile regression
lines(R$U,R$V, col=2, lwd=2)
R <- qua.regressCOP2(f=0.10, cop=PLACKETTcop, para=theta) # 10th-percentile regression
lines(R$U,R$V, col=2, lty=2)
mtext("Quantile Regression U wrt V for Plackett copula")#
## End(Not run)