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

Description

Perform quantile regression (Nelsen, 2006, pp. 217--218) using a copula by numerical derivatives of the copula (derCOPinv). If $X$ and $Y$ are random variables having quantile functions $x(F)$ and $y(G)$ and letting $y=\tilde{y}(x)$ denote a solution to $\mathrm{Pr}[Y \le y\mid X = x] = F$, where $F$ is a nonexceedance probability. Then the curve $y=\tilde{y}(x)$ is the quantile regression curve of $V$ or $Y$ with respect to $U$ or $X$, respectively. If $F=1/2$, then median regression is performed (med.regressCOP). Using copulas, the quantile regression is expressed as $$\mathrm{Pr}[Y \le y\mid X = x] = \mathrm{Pr}[V \le G(y) \mid U = F(x)] = \mathrm{Pr}[V \le v\mid U = v] = \frac{\delta \mathbf{C}(u,v)}{\delta u}\mbox{,}$$ where $v = G(y)$ and $u = F(x)$. The general algorithm is

Set $\delta \mathbf{C}(u,v)/\delta u = F$,
Solve the regression curve $v = \tilde{v}(u)$ (provided by derCOPinv), and
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.regressCOP(f=0.5, u=seq(0.01,0.99, by=0.01), cop=NULL, para=NULL, ...)

Value

An R

data.frame of the regressed probabilities of $V$ and provided $U=u$ values is returned.

Arguments

f: A single value of nonexceedance probability $F$ to perform regression at and defaults to median regression $F=1/2$;
u: Nonexceedance probability $u$ in the $X$ 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.

Author

W.H. Asquith

References