Compute the expected value of \(V\) given a \(U\) (the \(X\) direction) through the conditional distribution function \(F(X)\) using the appropriate partial derivative of a copula (\(\mathbf{C}(u,v)\)) with respect to \(U\). The inversion of the partial derivative is the conditional quantile function. Basic principles provide the expectation for a \(x \ge 0\) is
$$E[X] = \int_0^\infty xf(x)\mathrm{d}x = \int_0^\infty \bigl(1-F_x(X)\bigr)\mathrm{d}x\mbox{,}$$
which for the setting here becomes
$$E[V \mid U = u] = \int_0^1 \bigl(1 - \frac{\delta}{\delta u} \mathbf{C}(u,v)\bigr)\mathrm{d}v\mbox{.}$$
This function solves the integral using the derCOP function. Verification study is provided in the Note section.
EvuCOP(u=seq(0.01, 0.99, by=0.01), cop=NULL, para=NULL, asuv=FALSE, nsim=1E5,
subdivisions=100L, rel.tol=.Machine$double.eps^0.25, abs.tol=rel.tol, ...)Value(s) for the expectation are returned.
Nonexceedance probability \(u\) in the \(X\) direction;
A copula function with vectorization as in asCOP;
Vector of parameters or other data structures, if needed, to pass to the copula;
Return a data frame of the \(U\) and \(V\);
Number of simulations for Monte Carlo integration when the numerical integration fails (see Note in EvuCOP);
Argument of same name passed to integrate();
Argument of same name passed to integrate();
Argument of same name passed to integrate(); and
Additional arguments to pass to derCOP.
W.H. Asquith
EuvCOP, derCOP