copBasic (version 2.2.7)

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 $$\mathrm{Pr}[Y\le y\mid X=x]=\mathrm{Pr}[V\le G(y)\mid U=F(x)]=\mathrm{Pr}[V\le v\mid U=v]=\frac{\delta \mathbf{C}(u,v)}{\delta u}\text{,}$$ where $v=G(y)$ and $u=F(x)$. The general algorithm is

1. Set $\delta \mathbf{C}(u,v)/\delta u=F$,

2. Solve the regression curve $v=\tilde{v}(u)$ (provided by derCOPinv), 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.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

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

See Also

med.regressCOP, derCOPinv, qua.regressCOP.draw