Nelsen explained in his 1999 book *An introduction to copulas* how to fit a (nonlinear) quantile regression model by means of a bivariate copula (pp. 175-176).

In short, Nelsen’s method for fitting a (nonlinear) quantile regression model is as follows:

- Take the partial derivative of the copula function
*C(u, v)* with respect to *u*, where *u* and *v* are both defined in [0, 1]. Denote this partial derivative by *c*_{u}(u, v), and note that *c*_{u}(u, v) = P{V ≤ v | U = u}.
- For fitting the quantile regression model, set
*c*_{u}(u, v) = p, where *p* is defined in [0, 1].
- For regressing
*v* on *u*, evaluate *c*_{u}(u, v) at *u*, and subsequently solve *c*_{u}(u, v) = p for *v* (or, similarly, solve *0 = c*_{u}(u, v) – p for *v*).
- Replace
*u* by* F*_{x}^{-1}(u) and *v* by *F*_{y}^{-1}(v), where *F*_{x}^{-1}(·) and *F*_{y}^{-1}(·) are the quantile functions for *x* (=independent variable) and *y *(=dependent variable), respectively.

The following R code implements this copula method proposed by Nelsen for fitting a (nonlinear) quantile regression model. In addition, the R code may also compute confidence intervals for the fitted quantiles using a Monte-Carlo method. Continue reading R code for performing quantile regression using bivariate copulas